Lectures on Algebraic Rewriting

Philippe Malbos

Institut Camille Jordan Université Claude Bernard Lyon 1

- December 2019 - PHILIPPE MALBOS Univ Lyon, Université Claude Bernard Lyon 1 [email protected] CNRS UMR 5208, Institut Camille Jordan 43 blvd. du 11 novembre 1918 F-69622 Villeurbanne cedex, France

— December 6, 2019 - 17:47 — Contents

1 Abstract Rewriting 1 1.1 Abstract Rewriting Systems ...... 2 1.2 Conﬂuence ...... 4 1.3 Normalisation ...... 6 1.4 From local to global conﬂuence ...... 10

2 String Rewriting 13 2.1 Preliminaries: one and two-dimensional categories ...... 13 2.2 String rewriting systems ...... 21 2.3 The word problem ...... 25 2.4 Branchings ...... 26 2.5 Completion ...... 30 2.6 Existence of ﬁnite convergent presentations ...... 32

3 Coherent presentations and syzygies 35 3.1 Introduction ...... 35 3.2 Categorical preliminaries ...... 37 3.3 Coherent presentation of categories ...... 40 3.4 Finite derivation type ...... 42 3.5 Coherence from convergence ...... 44

4 Two-dimensional homological syzygies 51 4.1 Preliminaries on modules ...... 51 4.2 Monoids of ﬁnite homological type ...... 55 4.3 Squier’s homological theorem ...... 61 4.4 Homology of monoids with integral coefﬁcients ...... 63 4.5 Historical notes ...... 65

iii CONTENTS

5 Linear rewriting 69 5.1 Linear 2-polygraphs ...... 70 5.2 Linear rewriting steps ...... 77 5.3 Termination for linear 2-polygraphs ...... 79 5.4 Monomial orders ...... 80 5.5 Conﬂuence and convergence ...... 82 5.6 The Critical Branchings Theorem ...... 85 5.7 Coherent presentations of algebras ...... 90

6 Paradigms of linear rewriting 95 6.1 Composition Lemma ...... 96 6.2 Reduction operators ...... 98 6.3 Noncommutative Gröbner bases ...... 99

7 Anick’s resolution 103 7.1 Homology of an algebra ...... 104 7.2 Anick’s chains ...... 105 7.3 Anick’s resolution ...... 107 7.4 Computing homology with Anick’s resolution ...... 113 7.5 Minimality of Anick’s resolution ...... 115

Bibliography 118

iv CHAPTER 1

Abstract Rewriting

Contents 1.1 Abstract Rewriting Systems ...... 2 1.2 Conﬂuence ...... 4 1.3 Normalisation ...... 6 1.4 From local to global conﬂuence ...... 10

The principle of rewriting comes from combinatorial algebra. It was introduced by Thue when he considered systems of transformation rules on combinatorial objects such as strings, trees or graphs in order to solve the word problem, [Thu14]. Given a collection of objects and a system of transformation rules on these objects, the word problem is

INSTANCE: given two objects, QUESTION: can one of these objects be transformed to the other by means of a finite number of applications of the transformation rules? Dehn described the word problem for finitely presented groups, [Deh10] and Thue studied this problems for strings, which correspond to the word problem for finitely presented monoids, [Thu14]. Note that it was only much later, that the problem was shown to be undecidable, independently by Post [Pos47] and Markov [Mar47a, Mar47b]. Afterwards, the word problem have been considered in many contexts in algebra and in computer science. Far beyond the precursor works on this decidabilty problem on strings, rewriting theory has been mainly developed in theoretical computer science, producing numerous variants corresponding to different syntaxes of the formulas being transformed: strings in a monoid, [BO93, GM18], paths in a graph,

1 1.1. Abstract Rewriting Systems terms in an algebraic theory, [BN98, Klo92, Ter03], terms modulo, λ-terms, trees, Boolean circuits, [Laf03], graph grammars, etc. Rewriting appears also on various forms in algebra, for commutative algebras, [Buc65, Buc87], Lie algebras, [Shi62], with the notion of Gröbner-Shirshov bases, or associative algebras, [Bok76, Ber78, Mor94, Ufn95, GHM19] and operads, [DK10], as well as on topological objects, such as Reidemeister moves, knots or braids, [Bur01], or in higher-dimensional categories, [GM09, GM12a, Mim10, Mim14]. Many of the basic deﬁnitions and fundamental properties of these forms of rewriting can be stated on the most abstract version of rewriting that is given by a binary relation on set. In this chapter, we present the notion of abstract rewriting system and the main abstract rewriting properties used in these lectures. We refer the reader to [BN98, Klo92, Ter03] for a complete account on the abstract rewriting theory.

1.1. ABSTRACT REWRITING SYSTEMS

1.1.1. Abstract Rewriting Systems. An abstract rewriting system, ARS for short, is a data (A, I) made of a set A and a sequence I of binary relations on A indexed by a set I, that is, → →I = ( α)α∈I, and α ⊆ A × A.

The relation is called reduction→ or →rewrite relation on→A. An element (a, b) in the relation will be denoted by a b, and we said that b is a one-step reduct of a, and that a is a one-step expansion of b. An element of is called a reduction step. In most cases the elements of A have a syntactic→ or graphical nature→ (string, term, tree, graph, polynomial...). We will denoted by ≡ the syntactical or graphical identity. →

1.1.2. Reduction sequence. A reduction sequence, or rewriting sequence, with respect to a reduction relation is a ﬁnite or inﬁnite sequence of reduction steps

→ a0 a1 a2 ...

If we have a reduction sequence → → →

a ≡ a0 a1 a2 ... an−1 an ≡ b we say that a reduces to b. The length→of a→ ﬁnite reduction→ → sequence→ is the number of its reduction steps.

1.1.3. Composition. Given two reduction relations 1 and 2 on A, their composition is denoted by 1 · 2 and deﬁned by → → → → a 1 · 2 b if a 1 c 2 b, for some c in A.

1.1.4. Notations. The identity→ relation→ is denoted→ by→

0 = {(a, a) | a ∈ A}.

→

2 1.1.5. Branchings and conﬂuence pairs

The inverse relation of is denoted by , or by −, and deﬁned by:

= {(b, a) | a b}. → ← → 0 A relation is reflexive if ⊆ and transitive if · ⊆ . The reflexive closure of is denoted by ≡ ← → and defined by ≡ 0 → → =→ ∪→ →. → → The symmetric closure of is denoted by and defined by → → → = ∪ . → ↔ + The transitive closure of is denoted by and defined by ↔ → ← + [ i → → ⊆ , i>0 → → i+1 i ∗ where = · for all i > 0. The reflexive and transitive closure of is denoted by , or by , and defined by + 0 → → → = ∪ . → → ∗ The reflexive, transitive and symmetric closure of is denoted by and defined by → → ∗ = ( )∗. → ↔ ∗ In particular, we have a b is there is a rewriting↔ ↔ sequence from a to b and we have a b if and only if there is a zig-zag of rewriting sequence from a to b: ↔ a ≡ a0 a1 a2 ... an−1 an ≡ b.

∗ The relation is equal to the equivalence relation generated by . ↔ ↔ ↔ ↔ ↔

1.1.5. Branchings↔ and conﬂuence pairs. A branching (resp. →local branching) of the relation is an element of the composition · (resp. · ). It is deﬁned by a triple a c b (resp. a c b) as pictured by the following diagram: → ← → c c ← → (resp. ) a b a b

A confluence pair (resp. local confluence pair) of the relation is an element of the composition · (resp. · ). It is defined by a triple a d b as pictured by the following diagram: → a b a b → ← ÐÐ (resp. Ð ) d d

Note that the relations · and · are symmetric.

← →

3 1.2. Conﬂuence

1.1.6. Commutation. Two relations 1 and 2 on A commute if

→ 1 · →2 ⊆ 2 · 1 .

1.2. CONFLUENCE

1.2.1. Diamond property. A relation has the diamond property if it commutes with itself. This means that for any local branching a c b there exists a local conﬂuence: → ← → c a b Ð d

This property is hard to obtain in general. Let us give the main conﬂuence patterns used in rewriting.

1.2.2. Conﬂuence patterns. A reduction relation is called

∗ i) Church-Rosser if ⊆ · . →

ii) confluent if the relation↔ commutes, that is · ⊆ · . iii) semi-confluent if · ⊆ · . ≡ iv) strongly-confluent←if · ⊆ · .

v) locally conﬂuent if ←· →⊆ · ←. vi) has the diamond property← →if the relation commutes, that is · ⊆ · .

→ ← → → ← o ∗ / i) ii) iii)

≡ iv) v) vi)

4 1.2.3. Remark

1.2.3. Remark. The diamond property implies the Church-Rosser property, [New42, Theorem 1]. Note that in [New42] Newman called confluence the Church-Rosser property defined above. He showed that these properties coincide. Obviously, any Church-Rosser property is confluent, and the reverse implication is shown by the following diagram:

...

"" ||

|| ""

"" ||

1.2.4. Proposition. For an abstract rewriting system (A, ) the following conditions are equivalent

i) is conﬂuent, →

ii) → is semi-conﬂuent, iii) → has the Church-Rosser property.

Proof.→Prove that iii) implies i). Suppose that is Church-Rosser. Given a branching a c b, we ∗ have a b. Hence by the Church-Rosser property, there is a confluence pair a d b, hence is confluent. Obviously i) implies ii). Prove that ii) implies iii). Suppose that is semi-confluent and let ∗ → a b.↔ Prove by induction on the length of the sequence of reductions between a and b, that there→ is a confluence pair a d b. This is obvious when the sequence is of length 0→, that is a ≡ b, or when the n−1 1 sequence↔ is of length 1, that is a b or a b. Let consider a sequence of reductions a b0 b. 0 0 By induction hypothesis, there is a confluence pair a d b . If b b , that is → ← ↔ ↔ n − 1 1 → a o / b0 o b

Induction

& x & d x

0 by induction, this gives a conﬂuence pair a d b. In the other case, if b b, by semi-conﬂuence,

→

5 1.3. Normalisation

0 there is a conﬂuence pair d d b:

n − 1 1 a o / b0 / b

Semi- Induction Conﬂuence

%% yy d - - d0

0 hence, by induction we have a confluence pair a d b. It follows that the relation is Church- Rosser. → 1.2.5. Exercise. Prove that strong confluence implies confluence.

1.2.6. Exercise. Let A be a set and let 1 and 2 be two reduction relations on A. 1. Prove that the conﬂuence of 1 and 2 does not imply the conﬂuence of 1 ∪ 2. → → 2. Prove that → → → → 1 ⊆ 2 ⊆ 1 implies 1 = 2 .

3. Prove that if 1 ⊆ 2 ⊆→1 and→ 2 is strongly conﬂuent, then 1 is conﬂuent.

4. Prove that if 1 and 2 are conﬂuent and commute, then the relation 1 ∪ 2 is also conﬂuent. → → → → → → → → 1.3. NORMALISATION

Let (A, ) be an abstract rewriting system.

1.3.1. Normal→ form. An element a in A is in normal form, or irreductible, with respect to if there is no b in A such that a b. It is reductible if it is no irreductible. We denote by NF( ) the set of normal forms in A with respect to . → → → → 1.3.2. Normalizing. An element a in A is (weakly) normalizing if a b for some b in NF( ). Then we say that a has a normal form b and b is called a normal form of a. The relation is (weakly) normalizing if every element a in A is normalizing. → → 1.3.3. Termination. An element a in A is strongly normalizing if every reduction sequence starting from a is ﬁnite. The relation is strongly normalizing, or terminating, or noetherian if every a in A is

→

6 1.3.4. Convergence strongly normalizing. Any terminating relation is normalizing. Note that the converse is false as shown by the following abstract rewriting system

a0 / a1 / a2 / a3 / a4 / a5 / ···

} b qrstv

1.3.4. Convergence. We say that is convergent, or complete, canonical, uniquely terminating, if is confluent and terminating. → → 1.3.5. Normal form property. The relation has the normal form property if for any a in A and any normal form b in A ∗ a b →implies a b. The relation has the unique normal form property if for all normal forms a and b in A ↔∗ a b implies a ≡ b. → 1.3.6. Semi-convergence. We say that ↔ is semi-convergent, or semi-complete, if has the unique normal form property and is normalizing. If (A, ) is semi-convergent, then every element a in A reduces to a unique normal form denoted→ by ab. → → 1.3.7. Confluence and unicity of the normal form. If is confluent, every element has at most one normal form. As an immediate consequence of the equivalence of the Church-Rosser property and the confluence property, Proposition 1.2.4, we have → 1.3.8. Theorem. For an abstract rewriting system (A, ) the following implications hold: i) The normal form property implies the unique normal form property. → ii) If is confluent then has the normal form property. iii) If is semi-convergent then it is confluent. → → For a confluent abstract rewriting system (A, ), two elements a and b in A are equivalent if there are joignable:→ a · b. The test of joignability may be not possible when the relation is not terminating. For example, how to test the joignability→ of −n and n in the following example: ··· o −2 o −1 o 0 / 1 / 2 / ··· Let us show that normalisation suffices to determine joignability. If is normalizing and confluent, every element a in A has a unique normal form denoted by ab. 1.3.9. Theorem. If is normalizing and confluent then we have → ∗ a b if and only if ab ≡ b.b → As a consequence of the previous result, for a normalizing and confluent abstract rewriting system (A, ) an equivalence test of two↔ elements a and b in A is to check the syntactical equality of their normal forms ab and b. If the normal forms are computable and the syntactic identity is decidable then the equivalence→ is decidable.

7 1.3. Normalisation

1.3.10. Exercise. Prove Theorem 1.3.8 and Theorem 1.3.9.

1.3.11. Examples. The abstract rewriting system

' a g a0

Ò b is conﬂuent, not terminating and admits a unique normal form. The abstract rewriting system

' 0 co b / a f a is not conﬂuent, not terminating and admits a unique normal form.

1.3.12. Example. Let A = N − {0, 1}. Consider the relation on A deﬁned by the {(m, n) | m > n and n divides m }.

Then m is in NF( ) if and only if m is prime. An element p is a normal form of m if and only if p is a prime factor of m. We have m · n if and only if m and n are note relatively prime. The transitive closure of coincide woth because > and divide relations are already transitive. We have ∗ → = A × A and terminates and it is not confluent. → → ↔1.3.13. Example.→ Let A = {a, b}∗ be the free monoid on {a, b}. We consider the relation defined by the set {(ubav, uabv) | u, v ∈ A}. → n m Then an element of A is in normal form if and only if it is of the form a b for n, m ∈ N. Not that the relation terminates and its confluent. Thus every element of A has a unique normal form and we have 0 ∗ 0 0 w · w if and only if w w if and only if w and w contain the same number of as and bs. We will see→ in the next chapter on string rewriting that such an abstract rewriting system can be specified by only one rewriting rule ba ↔ab.

1.3.14. Exercise, [Jan88]. →Consider the set N × N with the reduction relation 1 deﬁned by (x, y) 1 (x0, y0) if

0 0 0 0 → → (x = x − 2) and (y = y > 1) or (x = x + 2) and (y = y − 1) .

1. Show that 1 is terminating.

2. Show that 1 is not confluent. → 3. Define a reduction relation 2 on N × N that is terminating, confluent and equivalent to 1, that ∗ ∗ is the relations →1 and 2 are equal. → → ↔ ↔

8 1.3.15. Well-founded induction

1.3.15. Well-founded induction. The principle of induction for natural numbers ensures that a property P(n) holds for all natural numbers n if we can show that P(n) holds under the hypothesis that P(m) holds for all m < n. The principle is a consequence of the fact that there is no inﬁnitely descending chain of natural numbers. The well-founded induction principle for an abstract rewriting system (A, ) can be stated as follows. Given a property P on elements of A, then → + ∀ a ∈ A, ∀ b ∈ A, a b implies P(b) implies P(a) implies → ∀a ∈ A, P(a).

With this principle, the property P(a) is proved for all elements a in A by proving that the property P(b) holds for any element b in A such that there is a rewriting sequence a b. 1.3.16. Theorem. If terminates then the principle of noetherian induction holds

Proof. Suppose that the→ principle of induction does not hold, that is + ∀ a ∈ A, ∀ b ∈ A, a b implies P(b) implies P(a)

0 holds and that there exist an element c in A such→ that P(c) does not hold. Then there exists c such that + c c0 and P(c0) does not hold. In this way, we construct an infinite reduction sequence starting on c. Hence, the reduction relation does not terminate. → Conversely, if the noetherian→ induction principe holds for an abstract rewriting system (A, ), then it terminates. It suffices to apply the induction principle to the property: → P(a) ≡ (there is no infinite reduction sequence starting on a) .

1.3.17. Exercise. Let (A, ) be an abstract rewriting system. The relation is called finitely branching if each element a of A has only finitely many direct successors, that is elements b such that a b. + The relation is called globally→ finite if the relation is finitely branching, that→ is each element a in A has only finitely many successors. → 1. Suppose that the relation is terminating and→ finitely branching. Prove that it is globally finite. 2. Show that it is not true that a finitely branching relation is terminating if it is globally finite. → + A relation is acyclic if there is no element a in A such that a a. 3. Show that any acyclic relation is terminating if it is globally finite. 4. Show that a finitely branching and acyclic relation is terminating→ if and only if it is globally finite.

1.3.18. Exercise. Let (A, ) be an abstract rewriting system such that every element a in A has a unique irreducible descendant. Prove that the relation is conﬂuent. → →

9 1.4. From local to global conﬂuence

1.4. FROM LOCAL TO GLOBAL CONFLUENCE

The local conﬂuence does not generally imply conﬂuence, however these properties are equivalent for terminating rewriting systems. This result is also due to Newman.

1.4.1. Theorem (Newman’s lemma, [New42, Theorem 3]). A terminating relation is conﬂuent if it is locally conﬂuent.

A short proof by Noetherian induction is given by Huet in [Hue80]. Due to this proof, Newman’s Lemma is also called the diamond lemma.

Proof. Suppose that is locally confluent and terminating. We prove its confluence by Noetherian induction. Given a0 in A, we suppose that for all a with a0 a and for all branching → / / a1 a / / a2 there exists a confluence a 1 >> t a2 Let us consider a branching / / a0

a0 / / a00

0 00 0 The cases a ≡ a0 or a ≡ a0 are obvious. In the other case, the length of the reductions a0 a and 00 a0 a are greater than 1:

1 a0 1 | 0 00 a1 loc. conﬂ. a1 ÒÒ 0 ' ' t t 00 a ind. hyp. a2 a

* * d0 s s ind. hyp.

w , , d w

We conclude using the induction hypothesis and local conﬂuence.

10 1.4.2. Example, [Hue80]

1.4.2. Example, [Hue80]. The following examples illustrate that the requirement of noetherianity is necessary to prove confluence from local confluence. The following abstract rewriting system is locally confluent but not confluent

# bo a a0 / b0 a

The following abstract rewriting system with 2n a, 2n + 1 b and n n + 1 for all n in N without cycle is local conﬂuent but not conﬂuent: → → → ;5 bO cik

5 1 ; 3 ; 5 ; 7 : 9 : 11

0 2 4 6 8 10 ···

# { ,) a u

It is locally conﬂuent but not conﬂuent.

11 1.4. From local to global conﬂuence

12 CHAPTER 2

String Rewriting

Contents 2.1 Preliminaries: one and two-dimensional categories ...... 13 2.2 String rewriting systems ...... 21 2.3 The word problem ...... 25 2.4 Branchings ...... 26 2.5 Completion ...... 30 2.6 Existence of ﬁnite convergent presentations ...... 32

A string rewriting system, SRS for short, historically called a semi-Thue system, is a rewriting system over a set of strings on an alphabet. String rewriting systems are Turing complete in the sense that they give a calculus that is equivalent to that of the Turing machine. String rewriting system appear in the formal language theory. They are also used in combinatorial algebra as a tool for presentation of semigroups, groups or monoids. For a fuller treatment on string rewriting systems we refer the reader to [BO93] and [Jan88]. In this chapter, string rewriting system will be describe in the categorical language of 2-polygraphs as in [GM18] and [GM12b, Section 4]. A 2-polygraph is a rewriting system over a set of paths of a given directed graph. String rewriting system is the particular case when the directed graph has only one vertex.

2.1. PRELIMINARIES: ONEANDTWO-DIMENSIONAL CATEGORIES

13 2.1. Preliminaries: one and two-dimensional categories

2.1.1. Categories. A (small) category (or 1-category) is a data C made of

i) a set C0, whose elements are called the 0-cells of C,

ii) for every 0-cells x and y of C, a set C(x, y), whose elements are called the 1-cells from x to y of C, iii) for every 0-cells x, y and z of C, a map

x,y,z ?0 : C(x, y) × C(y, z) C(x, z),

called the composition (or 0-composition) of C, → iv) for every 0-cell x, a speciﬁed element 1x of C(x, x), called the identity of x.

The following relations are required to hold

v) the composition is associative, i.e., for every 0-cells x, y, z and t and for every 1-cells u ∈ C(x, y), v ∈ C(y, z) and w ∈ C(z, t),

x,z,t x,y,z x,y,t y,z,t ?0 (?0 (u, v), w) = ?0 (u, ?0 (v, w)), vi) the identities are local units for the composition, i.e., for every 0-cells x and y and for every 1-cell u ∈ C(x, y), x,x,y x,y,y ?0 (1x, u) = u = ?0 (u, 1y).

We write u : x y to mean that u is in C(x, y). The 0-cell x is the source of u denoted by s0(u) x,y,z and the 0-cell y is the target of u denoted by t0(u). The composition ?0 (u, v) will be denoted by u ?0 v, or simply by→ juxtaposition uv.

2.1.2. Monoids. A monoid M with product · and identity element 1M corresponds to a category M with only one 0-cell, denoted by •, and the 1-cells of M(•, •) are the elements of the monoid M. The identity arrow 1• of M is the identity element 1M and the composition of u ?0 v of 1-cells in M(•, •) if the product u · v in the monoid M. The associativity and unitary properties of the composition, making M into a category, are induced by the corresponding properties of the product ·. In this way, any monoid can be thought of as a one-0-cell category and a category can be thought of as a "monoid with several 0-cells".

2.1.3. Internal definition. A category C can also be defined as an internal category in the category Set of sets. Explicitly, it is defined by a diagram in Set:

t0 oo ? s o 0 × C0 0 / C1 C1 C0 C1 i1

14 2.1.4. Product of categories

where C1 ×C0 C1 is deﬁned by the following pullback diagram in the category Set: / C1 ×C0 C1 C1

s0 C1 / C0 t0

Elements of C1 ×C0 C1 are pairs (u, v) of 0-composable 1-cells u and v, that is satisfying t0(u) = s0(v). The maps s0, t0 and ?0 satisfy the axioms in such a way that the diagram above deﬁnes a category. Explicitly, the following diagrams commute in the category Set:

i i ? ? 1 / 1 / 0 / 0 / C0 C1 C0 C1 C1 ×C0 C1 C1 C1 ×C0 C1 C1

s0 t π s0 π t id id 0 1 2 0 C0 C0 C1 / C0 C1 / C0 s0 t0

? × id i × id id × i 0 C0 / 1 C0 / o C0 1 C1 ×C0 C1 ×C0 C1 C1 ×C0 C1 C0 ×C0 C1 C1 ×C0 C1 C1 ×C0 C0 ? id ×C0 ?0 ?0 0 π2 π1 ' w C1 ×C C1 / C1 C0 0 ?0 where π1 and π2 denote respectively ﬁrst and second projection.

2.1.4. Product of categories. Given two categories C and D, the product category C × D is deﬁned as follows i) the 0-cells are the pairs (x, y), where x is a 0-cell of C and y is a 0-cell of D,

ii) the 1-cells are the pairs (u, v) where u is a 1-cell of C and v is a 1-cell of D, iii) the composition is component-wise: (u, v)(u0, v0) = (uu0, vv0), iv) the identities are the pairs of identities: 1(x,y) = (1x, 1y).

2.1.5. Functors. Let C and D be categories. A functor F : C D is a data made of

i) a map F0 : C0 D0, → ii) for every 0-cells x and y of C, a map → Fx,y : C(x, y) D(F(x),F(y)), such that the following relations are satisﬁed: →

15 2.1. Preliminaries: one and two-dimensional categories iii) for every 0-cells x, y and z and every 1-cells u : x y and v : y z of C,

F (u ? v) = F (u) ? F (v), x,z 0 x,y→ 0 y,z → iv) for every 0-cell x of C, Fx,x(1x) = 1F(x).

We will write F(x) for F0(x) and F(u) for Fx,y(u). A functor F is a monomorphism (resp. an epimorphism, resp. an isomorphism) if the map F0 and each map Fx,y is an injection (resp. a surjection, resp. a bijection).

2.1.6. Functors as morphisms of graphs. A functor F : C D can be seen as a morphism of graphs

s o 0 → C0 o C1 t0 F0 F1 s o 0 D0 o D1 t0 where, for every 1-cell u : x y of C, the 1-cell F1(u) is deﬁned as Fx,y(u).

2.1.7. One-dimensional polygraphs.→ A 1-polygraph is a directed graph Σ, i.e., a diagram of sets and maps s o 0 Σ0 o Σ1. t0

The elements of Σ0 and Σ1 are called the 0-cells and the 1-cells of Σ, respectively. If there is no confusion, we just write Σ = (Σ0,Σ1). Note that the notion of 1-polygraph is equivalent to the notion of abstract rewriting system given in (1.1.1). A 1-polygraph is ﬁnite if it has ﬁnitely many 0-cells and 1-cells.

∗ 2.1.8. Free categories. If Σ is a 1-polygraph, the free category over Σ is the category denoted by Σ1 and deﬁned as follows:

∗ i) the 0-cells of Σ1 are the ones of Σ, ∗ ii) the 1-cells of Σ1 from x to y are the ﬁnite paths of Σ, i.e., the ﬁnite sequences

u1 u2 u3 un−1 un x / x1 / x2 / ··· / xn−1 / y

of 1-cells of Σ, iii) the composition is given by concatenation, iv) the identities are the empty paths.

∗ If Σ has only one 0-cell, then the 1-cells of the free category Σ1 form the free monoid over the set Σ1.

16 2.1.9. Generating 1-polygraph

2.1.9. Generating 1-polygraph. Let C be a category. A 1-polygraph Σ generates C if there exists an epimorphism ∗ π : Σ1 − C that is the identity on 0-cells. In that case, the 1-polygraph Σ has the same 0-cells as C and, for every → 0-cells x and y of C, the map ∗ π : Σ1(x, y) − C(x, y) is surjective. A category is ﬁnitely generated if it admits a ﬁnite generating 1-polygraph. →

2.1.10. Spheres and cellular extensions of categories. A sphere of a category C is a pair γ = (u, v) of parallel 1-cells of C, that is, with the same source, s0(u) = s0(v), and the same target, t0(u) = t0(v). The 1-cell u is the source of γ and v is its target.A cellular extension of C is a set Γ equipped with a map from Γ to the set of spheres of C. It is equivalent to the data of a set Γ with two maps

s1 C oo Γ. t1 satisfying the following gobular relations:

s0s1 = s0t1, t0s1 = t0t1.

An element of Γ will be graphically represented by a 2-cell with the following globular shape

f p α q Õ @ g

α that relates parallel 1-cells u and v in C, also denoted by u v or by α : u v.

2.1.11. Congruences. A congruence on a category C is an⇒ equivalence relation⇒ ≡ on the parallel 1-cells of C that is compatible with the composition of C, that is, for every 1-cells

u w ! w0 x / y = z / t v of C such that u ≡ v, we have wuw0 ≡ wvw0. If Γ is a cellular extension of C, the (Thue) congruence generated by Γ is denoted by ≡Γ and deﬁned as the smallest congruence relation such that, if γ is in Γ, then s1(γ) ≡Γ t1(γ).

17 2.1. Preliminaries: one and two-dimensional categories

2.1.12. Quotient categories. If C is a category and Γ is a cellular extension of C, the quotient of C by Γ is the category denoted by C/Γ and deﬁned as follows: i) the 0-cells of C/Γ are the ones of C, ii) for every 0-cells x and y of C, the set C/Γ(x, y) of 1-cell with source x and target y is the quotient of C(x, y) by the restriction of ≡Γ . We will denote by πΓ : C − C/Γ Γ the canonical projection. We will denote by u for the image through πΓ of a 1-cell u in C. The superscript Γ in uΓ will be omitted whenever ambiguity→ is not introduced.

2.1.13. Two-dimensional categories. A (strict) 2-category is a category enriched over the cartesian monoidal category Cat of categories. Explicitly, is a data C made of a set C0, whose elements are called the 0-cells of C, and, for every 0-cells x and y of C, a category C(x, y), whose 0-cells and 1-cells are respectively called the 1-cells and the 2-cells from x to y of C. This data is equipped with the following algebraic structure: i) for every 0-cells x, y and z of C, a functor x,y,z ?0 : C(x, y) × C(y, z) C(x, z),

ii) for every 0-cell x, a speciﬁed 0-cell 1x of the category C(x,→ x). The following relations are required to hold: iii) the composition is associative, i.e., for every 0-cells x, y, z and t, x,z,t x,y,z x,y,t y,z,t ?0 ◦ (?0 × IdC(z,t)) = ?0 ◦ (IdC(x,y) ×?0 ), iv) the identities are local units for the composition, i.e., for every 0-cells x and y, x,x,y x,y,y ?0 ◦ (1x × IdC(x,y)) = IdC(x,y) = ?0 ◦ (IdC(x,y), 1y).

2.1.14. Globular definition. A 2-category can, equivalently, be defined as a 2-graph s s o 0 o 1 C0 o C1 o C2 t0 t1 equipped with an additional algebraic structure. The definition of 2-graph requires that the source and target maps satisfy the globular relations:

s0 ◦ s1 = s0 ◦ t1 and t0 ◦ s1 = t0 ◦ t1.

The 2-graph is equipped with two compositions, the 0-composition ?0 and the 1-composition ?1, respectively deﬁned on 0-composable 1-cells and 2-cells, and on 1-composable 2-cells. We also have an inclusion of C0 into C1 given by the identities of the 2-category, and an inclusion of C1 into C2 induced by the identities of the hom-categories. Explicitly, we have the following operations:

18 2.1.14. Globular deﬁnition

u v i) for every 1-cells x − y − z, a 0-composite 1-cell

u ? v : x z, → → 0

u v → ii) for every 2-cells x f y g B z , a 0-composite 2-cell Õ @ Õ u0 v0

u ?0 v x f ?0 g ? z , Õ 0 0 u ?0 v

f iii) for every 2-cells x v Õ / y , a 1-composite 2-cell g D Õ w

u x f ?1 g > y , Õ w iv) for every 0-cell x, an identity 1-cell 1x : x x, u v) for every 1-cell x − y, an identity 2-cell → 1 : u u. → u

The 0-composition and the 1-composition satisfy the following→ relations: u v w − for every 1-cells x − y − z − t, (u ?0 v) ?0 w = u ?0 (v ?0 w), u − for every 1-cell x − →y, 1→x ?0 u→= u = u ?0 1y, u v − for every 1-cells x − y − z, 1 = 1 ? 1 , → u?0v u 0 v f g h → → − for every 2-cells u %9 v %9 w %9 x, (f ?1 g) ?1 h = f ?1 (g ?1 h),

19 2.1. Preliminaries: one and two-dimensional categories

u v w − for every 2-cells x f y g B z h B t , (f ?0 g) ?0 h = f ?0 (g ?0 h), Õ @ Õ Õ u0 v0 w0

u − for every 2-cell x f y , 1x ?0 f = f = f ?0 1y, Õ @ v

f − for every 2-cell u %9 v, 1u ?1 f = f = f ?1 1v,

u v

f g − for every 2-cells x 0 Õ / y 0 Õ / z , (f ? f0) ? (g ? g0) = (f ? g) ? (f0 ? g0). u B v C 1 0 1 0 1 0 f0 g0 Õ Õ u00 v00

The last relation is usually called the exchange relation or the interchange law for the compositions ?0 and ?1. This globular deﬁnition of 2-categories is equivalent to the enriched one. The 0-composition of 2-cells with identity 1-cells deﬁnes the whiskering operations:

u w ?0 u

w / − for every x y f @ z , the left whiskering is x w ?0 f ? z , Õ Õ v w ?0 v

u u ?0 w

w / − for every x f ? y z , the right whiskering is x f ?0 w ? z . Õ Õ v v ?0 w

The left and right whiskering operations satisfy the following relations:

f u / 0 Õ / 0 0 − for every x y v Dz , u ?0 (f ?1 f ) = (u ?0 f) ?1 (u ?0 f ), f0 Õ v00

20 2.2. String rewriting systems

u f v − for every x 0 Õ / y / z , (f ? f0) ? v = (f ? v) ? (f0 ? v), u C 1 0 0 1 0 f0 Õ u00 w u / v / − for every x y z f A t , (u ?0 v) ?0 f = u ?0 (v ?0 f), Õ w0 v

u / w / − for every x y f @ z t , (u ?0 f) ?0 w = u ?0 (f ?0 w), Õ v0 u

v / w / − for every x f ? y z t , (f ?0 v) ?0 w = f ?0 (v ?0 w), Õ u0

As for categories, we usually write uv and fg instead of u ?0 v and f ?0 g.

2.2. STRINGREWRITINGSYSTEMS

2.2.1. Two-dimensional polygraphs. A 2-polygraph is a triple Σ = (Σ0,Σ1,Σ2) made of a 1-polygraph ∗ (Σ0,Σ1), often simply denoted by Σ1, and a cellular extension Σ2 of the free category Σ1. In other terms, a 2-polygraph Σ is a 2-graph s s o 0 ∗ o 1 Σ0 o Σ1 o Σ2 t0 t1 whose 0-cells and 1-cells form a free category. The elements of Σk are called the k-cells of Σ and Σ is ﬁnite if it has ﬁnitely many cells in every dimension.

2.2.2. Example. The string rewriting system on the alphabet {a} with only one rewriting rule aa a is described by the 2-polygraph Σ, where α → Σ0 = {•},Σ1 = {a},Σ2 = {aa a}. The rule aa a corresponds to the following globular 2-cell ⇒ a 4 • a → α • Õ 4 • a

21 2.2. String rewriting systems

2.2.3. Presentations of categories. If Σ is a 2-polygraph, the category presented by Σ is the category denoted by Σ and deﬁned by ∗ Σ = Σ1/Σ2. If C is a category, a presentation of C is a 2-polygraph Σ such that C is isomorphic to Σ. In that case, the 1-cells of Σ are the generators of C, and the 2-cells of Σ are the relations of C. Two 2-polygraphs Σ and Υ are said to be Tietze-equivalent if they present isomorphic categories, that is there exists an isomorphism of categories Σ ' Υ.

2.2.4. Free 2-categories. Let Σ be a 2-polygraph. The free 2-category over Σ is the 2-category denoted ∗ by Σ2 and deﬁned as follows: ∗ i) the 0-cells of Σ2 are the ones of Σ, ∗ ii) for every 0-cells x and y of Σ, the category Σ2(x, y) is deﬁned as − the free category over the 1-polygraph ∗ – whose 0-cells are the 1-cells in Σ1(x, y), – whose 1-cells are the u w ! w0 x / y α z / t Õ = v

0 ∗ with α : u v in Σ2 and w and w in Σ1, − quotiented by the congruence generated by the cellular extension made of all the possible

⇒ 0 0 αwv ?1 u wβ ≡ uwβ ?1 αwv ,

0 0 ∗ for α : u u and β : v v in Σ2 and w in Σ1: u v ⇒ w ⇒ v u w α @ / / / / β @ Õ Õ u0 v0 ?1 = ?1 v u u0 w w v0 / / β @ α @ / / Õ Õ v0 u0 iii) for every 0-cells x, y and z of Σ the composition functor is given by the concatenation on 1-cells and, on 2-cells, as follows: 0 0 0 0 u1α1u1 ?1 ··· ?1 umαmum ?0 v1β1v1 ?1 ··· ?1 vnβnvn 0 0 0 0 = u1α1u1v1s(β1)v1 ?1 ··· ?1 umαmumv1s(β1)v1 0 0 0 0 ?1 umt(αm)umv1β1v1 ?1 ··· ?1 umt(αm)umvnβnvn

22 2.2.5. Rewriting sequences

0 0 u1 u1 v1 s(β1) v1 / α1 C / / / / Õ

. . 0 0 u1 u1 v1 v1 / α1 C / / β1 C / Õ Õ ?1 u0 s(β ) v0 ?1 ?1 um m v1 1 1 / αmC / / / / Õ . . . ?0 . ≡ ?1 t(α ) u0 v0 ?1 ?1 um m m v1 1 / / / / β1 C / 0 0 Õ um um vn vn / αmC / / βnC / Õ Õ ?1

. .

0 0 um t(αm) um vn vn / / / / βnC / Õ

∗ iv) for every 0-cell x, the identity 1-cell 1x is the one of Σ1.

∗ ∗ By deﬁnition of the 2-category Σ2, for every 1-cells u and v of Σ1, we have u = v in the quotient ∗ category Σ if, and only if, there exists a “zig-zag” sequence of 2-cells of Σ2 between them:

f1 g1 f2 gn−1 fn gn u %9 u1 ey v1 %9 u2 ey (··· ) %9 un−1 ey vn−1 %9 un ey v.

∗ 2.2.5. Rewriting sequences. A rewriting step of a 2-polygraph Σ is a 2-cell of the free 2-category Σ2 with shape l u ! v x / y ϕ z / t Õ = r

∗ where ϕ : l r is a generating 2-cell in Σ and u and v are 1-cells of Σ1. Such a rewriting step will be denoted by ulv Σ2 urv. The subscript Σ2 will be omitted whenever ambiguity is not introduced. A rewriting⇒ sequence of Σ is a ﬁnite or inﬁnite sequence ⇒

u1 Σ2 u2 Σ2 ... Σ2 un Σ2 ...

0 0 of rewriting steps. If Σ has a non-empty⇒ rewriting⇒ sequence⇒ from w⇒to w , we say that w rewrites into w . ∗ Let us note that every 2-cell f of the 2-category Σ2 decomposes into a ﬁnite rewriting sequence of Σ, this decomposition being unique up to exchange relations.

23 2.2. String rewriting systems

2.2.6. Leftmost reduction. Let Σ be a 2-polygraph. A reduction step w w0 is leftmost, and we denote w ` w0, if the two following conditions are satisﬁed

0 ∗ ⇒ i) if w = ulv and w = urv for some l r in Σ2 with u and v in Σ , ⇒ 1 ii) for any factorisation w = u0l0r0 for some l0 r0 in Σ , then ul is a proper prefix of u0l0 or ⇒ 2 ul = u0l0 and u is a prefix of u0. ⇒ 2.2.7. Rewriting properties of 2-polygraphs. To any 2-polygraph Σ, we associate an abstract rewriting ∗ system whose elements are 1-cells in Σ1 and the reduction relations is the relation Σ2 . We say that a 2- polygraph has a rewriting property P, such as normalisation, termination or confluence, if the associated ∗ abstract rewriting system (Σ1, Σ2 ) has the property P. In particular, a 2-polygraph⇒ is confluent if and only if it is Church-Rosser and by Newman’s lemma, Theorem 5.5.12, for a terminating 2-polygraph, local confluence and confluence⇒ are equivalent properties. nf ∗ We will denote by Σ1 the set of 1-cells of Σ1 in normal form with respect to Σ2. 2.2.8. Theorem. Let Σ be a terminating 2-polygraph. Then Σ is confluent if and only if the restriction of the canonical projection ∗ π : Σ1 − Σ to the irreducible 1-cells induces a bijection for any 0-cells x and y: → nf ∼ πex,y : Σ1 (x, y) − Σ(x, y).

2.2.9. Exercise. Prove Theorem 2.2.8. →

∗ 2.2.10. Termination order. A termination order on Σ is an order relation ≺ on parallel 1-cells of Σ1 such that the following three conditions are satisﬁed:

∗ 0 0 i) the composition of 1-cells of Σ1 is strictly monotone in both arguments, i.e., u ≺ u implies vu w ≺ 0 ∗ vuw for all composable 1-cells u, u , v and w in Σ1, ∗ ii) the relation is wellfounded, i.e., every decreasing family (un)n∈N of parallel 1-cells of Σ1 is station- ary, iii) for every 2-cell α of Σ2, the strict inequality t(α) ≺ s(α) holds. As a direct consequence of the deﬁnition, if a 2-polygraph admits a termination order, then it terminates.

2.2.11. Lexicographic order. A useful example of termination order is the left degree-wise lexicographic order (or deglex for short) generated by a given order on the 1-cells of Σ. It is deﬁned by the following strict inequalities, where each xi and yj is a 1-cell of Σ:

x1 ··· xp < y1 ··· yq, if p < q,

x1 ··· xk−1xk ··· xp < x1 ··· xk−1yk ··· yp, if xk < yk.

The deglex order is total if and only if the original order on the set Σ1 is total.

24 2.2.12. Reduced 2-polygraph

2.2.12. Reduced 2-polygraph. A 2-polygraph Σ is

i) left-reduced if for any l r in Σ2 then l is irreducible with respect to Σ2 \{l r},

ii) right-reduced if for any l r in Σ2 then r is irreducible with respect to Σ2, ⇒ ⇒ iii) reduced if it is left-reduced and right-reduced. ⇒ 2.2.13. Exercise [Mét83], [Squ87, Theorem 2.4]. Show that every ﬁnite convergent 2-polygraph is Tietze equivalent to a ﬁnite reduced convergent 2-polygraph.

2.3. THEWORDPROBLEM

2.3.1. The word problem. The word problem for a 2-polygraph Σ is the following decision problem ∗ INSTANCE: two 1-cells u and v in Σ1. ∗ QUESTION: Does u Σ2 v ? There are finite string rewriting systems for which the word problem is algorithmically unsolvable. Hence, the word problem⇐⇒ for finite string rewriting systems is undecidable in general. When a string rewriting system is finite and convergent, then its word problem is decidable by the normal form procedure.

∗ 2.3.2. Normal form procedure. Given a convergent 2-polygraph Σ, every 1-cell u of Σ1 has a unique ∗ normal form, denoted by ub, so that we have u = v in Σ if, and only if, ub = bv holds in Σ1. This deﬁnes a section ∗ Σ Σ1 ∗ of the canonical projection Σ1 Σ, mapping a 1-cell u of Σ to the unique normal form of its represen- ∗ tative 1-cells in Σ1, still denoted by ub. As a consequence, a ﬁnite and convergent 2-polygraph Σ yields a decision procedure for the word problem of the category Σ it presents: the normal-form procedure:

∗ Input: u, v two 1-cells of Σ1. begin reduce u to its normal form ub with respect to Σ2 ; reduce v to its normal form bv with respect to Σ2 ; if ub = bv then Accept else Reject end end Algorithm 1: Normal form procedure

Note that ﬁniteness is used to test whether a given 1-cell u is a normal form or not, by examination of all the relations and their possible applications on u. Then, the equality u = v holds in Σ if, and only ∗ if, the equality ub = bv holds in Σ1.

25 2.4. Branchings

2.3.3. Complexity of the word problem for a finite 2-polygraph. For a finite convergent 2-polygraph ∗ Σ, consider a function fΣ : N N such that for any 1-cell u in Σ1, the leftmost reduction sequence from u to its normal form contains at most fΣ(`(u)) many steps. In [Boo82], Book proves that for a finite ∗ convergent and reduced 2-polygraph→ Σ, the normal form for a 1-cell u in Σ1 can be computed in time O(`(u)+fΣ(`(u)). As a consequence, if a 2-polygraph Σ is length-reducing and confluent, then its word problem is decidable in linear time.

2.3.4. Decidability of the word problem and Tietze invariance. It is well-known that the decidability of the word problem is an invariant property of finite presentations of monoids: 2.3.5. Proposition. Let Σ and Υ two finite Tietze-equivalent 2-polygraphs. Then the word problem for Σ is decidable if and only if the word problem for Υ is decidable. We can thus talk about the decidability of the word problem in a finitely generated monoid. Fi- nally, let us mention the following result obtained by Avenhaus and Madlener in [AM78a, AM78b] for presentations of groups, but the proof can be applied to presentation of monoids. 2.3.6. Theorem. Let Σ and Υ be two Tietze-equivalent finite 2-polygraphs. If the word problem can be decided for Σ in time O(f(n)), then the word problem for Υ can be solved in time O(f(c.n)) for some constant natural number c > 0.

2.4. BRANCHINGS

∗ 2.4.1. Branchings. Recall from (1.1.5), that a branching of Σ is a pair (f, g) of 2-cells of Σ2 with a common source, as in the following diagram

f &: v u

g #7 w

The 1-cell u is the source of this branching and the pair (v, w) is its target. A branching (f, g) is local if f and g are rewriting steps. A branching

f &: v u

g #7 w

0 0 ∗ is conﬂuent if there exist 2-cells f and g in Σ2, as in the following diagram:

0 f '; v f + u u0 5I g #7 w g0

26 2.4.2. Local branchings

2.4.2. Local branchings. Local branchings belong to one of the three following families. The aspherical branchings have shape f u + v 2F f where f is a rewriting step. The orthogonal branchings, also called Peiffer branchings, have shape

0 fv (< u v

ug "6 uv0 where f : u u0 and g : v v0 are rewriting steps. The overlapping branchings are the remaining local branchings. ⇒ ⇒ 2.4.3. Critical branchings. Local branchings are compared by the order v generated by the relations

(f, g) v ufv, ugv)

∗ given for any local branching (f, g) and any possible 1-cells u and v of Σ1. An overlapping local branching that is minimal for the order v is called a critical branching, or a critical pair. Note that a 2-polygraph has two kinds of critical branchings, namely inclusion ones and overlapping ones, respectively corresponding to the two situations pictured on Figure 2.4.3:

EY EY / / / / / / I D Õ Õ

Figure 2.4.3: Critical branchings by inclusion and overlapping

2.4.4. Theorem (Critical pair theorem). A 2-polygraph is locally conﬂuent if, and only if, all its critical branchings are conﬂuent.

Proof. Every aspherical branching is conﬂuent:

f &: v 1v u ( v 5I

f $8 v 1v

27 2.4. Branchings

We also have conﬂuence of every Peiffer local branching:

0 0 fv (< u v u g - uv u0v0 3G ug "6 uv0 fv0

Finally, in the case of an overlapping but not minimal local branching (f, g), there exist factorisations f = uhv and g = ukv with

h '; w1 w

k #7 w2 a critical branching of Σ. By hypothesis, the branching (h, k) is conﬂuent:

0 h (< w1 h , w w0 4H 0 k "6 w2 k then so is (f, g): 0 f (< uw1v uh v / uwv uw0v 0D g 0 "6 uw2v uk v

2.4.5. Example. Consider the 2-polygraph Σ, with Σ1 = {a, b} and Σ2 = {α : aba 1}. The 2- polygraph Σ is terminating since `(u) > `(v) whenever u v. The polygraphs admits one critical branching and this branching is not conﬂuent: ⇒ ⇒ abα '; ab

ababa

αba #7 ba

It follows that the 2-polygraph Σ is not conﬂuent.

28 2.4.6. Example

2.4.6. Example. Consider the 2-polygraph Σ, with Σ1 = {a, b, c} and

Σ2 = {α : ab ca, β : bc ab, γ : ca bc}.

The 2-polygraph Σ is not terminating and⇒ local conﬂuent⇒ with three conﬂuent⇒ critical branchings:

aα cγ aβ )= aab %9 aca aγ cα )= cca %9 cbc cβ * * abc abc cab cab 6J 6J αc !5 cac %9 bcc βc γb !5 bcb %9 abb αb γc βb

bβ bγ )= bbc %9 bab bα * bca bca 6J βa !5 γa aba αa %9 caa

2.4.7. Example: reduced standard presentation. Given a category C, we call reduced standard polygraphic presentation of C, the 2-polygraph Σ deﬁned as follows:

i) it has one 0-cell for each 0-cell of C and one 1-cell ub : x y for every non-identity 1-cell u : x y of C, → → ii) it has one 2-cell 7 y ub bv µ u,v x Õ 3 z uvc for every non-identity 1-cells u : x y and v : y z of C such that uv is not an identity, iii) it has one 2-cell → → 7 y ub bv µ u,v x Õ x 1x

for every non-identity 1-cells u : x y and v : y x of C such that uv = 1x.

The 2-polygraph Σ is reduced and convergent.→ It has→ one critical branching (µu,vw,b uµb v,w) for every triple (u, v, w) of non-identity composable 1-cells in C. Each of these critical branchings is conﬂuent,

29 2.5. Completion with four possible cases, depending on whether uv or vw is an identity or not:

µu,vw uvw µuv,w µu,vw '; w b )= c b b b Th , uvw uvw uvw µu,vw bbb d3G bbb

!5 µu,vw "6 uvw uµb v,w ubvwc uµb v,w b c

µu,vwb µu,vwb (< uvcwb , µ uvw u = w ubbvwb uv,w bbb b 4H b w u uµv,w uµb v,w $8 b b

Following Theorem 2.4.4, one can decide whether a finite string rewriting system is convergent by checking confluence of critical branchings. If the set of rules is finite, there are only finitely many critical branchings. It thus can be tested whether every such branching is confluent. The result follows because, the rewriting system is locally confluent if and only if every critical branching is confluent.

2.4.8. Theorem ([Niv73]). Let Σ be a finite terminating string rewriting system. Then, whether or not Σ is locally confluent, is decidable. Hence, it is decidable whether or not Σ is confluent.

2.5. COMPLETION

2.5.1. Knuth-Bendix’s completion procedure. Let Σ be a terminating 2-polygraph, equipped with a total termination order ≺.A Knuth-Bendix’s completion of Σ is a 2-polygraph KB(Σ) obtained by the following procedure.

30 2.5.1. Knuth-Bendix’s completion procedure

Input: Σ be a terminating 2-polygraph with a total termination order ≺. KB(Σ) Σ Cb { critical branchings of Σ } while C←b 6= ∅ do ←Picks a branching in Cb: f $8 v u g &: w Cb Cb \{(f, g)} Reduce v to a normal form bv with respect to KB(Σ)2 Reduce← w to a normal form wb with respect to KB(Σ)2 f $8 v %9 bv u g %9 w %9 wb

if bv 6= wb then if bv > wb then KB(Σ)2 KB(Σ)2 ∪ { α : bv wb }: f ← ⇒ $8 v %9 bv u α Õ g &: w %9 wb

end if wb > bv then KB(Σ)2 KB(Σ)2 ∪ { α : wb bv }: f ← ⇒ $8 v %9 v bEY u α g &: w %9 wb

end end Cb Cb ∪ { critical branching created by α } end ← Algorithm 2: Knuth-Bendix completion procedure

If the procedure stops, it returns the 2-polygraph KB(Σ). Otherwise, it builds an increasing sequence of 2-polygraphs, whose limit is denoted by KB(Σ). Note that, if the starting 2-polygraph Σ is already convergent, then the Knuth-Bendix’s completion of Σ is Σ.

31 2.6. Existence of ﬁnite convergent presentations

2.5.2. Theorem ([KB70]). The Knuth-Bendix’s completion KB(Σ) of a 2-polygraph Σ is a convergent presentation of the category Σ. Moreover, the 2-polygraph KB(Σ) is ﬁnite if, and only if, the 2-polygraph Σ is ﬁnite and if the Knuth-Bendix’s completion procedure halts.

2.5.3. Exercice. Find a ﬁnite convergent presentation of the monoid generated by two generators a and b and submitted to the relation aba = 1.

2.6. EXISTENCE OF FINITE CONVERGENT PRESENTATIONS

When a string rewriting system is not convergent, one wishes to determine whether there exists a Tietze equivalent convergent string rewriting system. We can formulate the two following problems of existence of ﬁnite convergent presentations.

2.6.1. Problem.

INSTANCE: A ﬁnite string rewriting system (Σ1,Σ2).

QUESTION: Does (Σ1,Σ2) is Tietze equivalent to a ﬁnite convergent string rewriting system (Σ1,Υ2) ?

2.6.2. Problem.

INSTANCE: A ﬁnite string rewriting system (Σ1,Σ2).

QUESTION: Does (Σ1,Σ2) is Tietze equivalent to a ﬁnite convergent string rewriting system ? 2.6.3. Theorem ([BO84]). The problems 2.6.1 and 2.6.2 are undecidable.

2.6.4. Existence of ﬁnite convergent presentations. The normal form procedure proves that, if a monoid admits a ﬁnite convergent presentation, then it has a decidable word problem. The converse implication was still an open problem in the middle of the eighties. Jantzen in [Jan82, Jan85] asked the following question.

2.6.5. Question. Does every ﬁnitely presented monoid with a decidable word problem admit a ﬁnite convergent presentation?

+ 2.6.6. Example. In [KN85], Kapur and Narendran consider Artin’s presentation of the monoid B3 of positive braids on three strands:

Σ = s, t sts tst . The generators s and t correspond to the following braids ⇒ s = and t = and the rule sts tst corresponds to the Yang-Baxter relation:

⇒ = .

32 2.6.9. Question

+ They proved that the word problem for B3 is decidable and that this monoid admits no ﬁnite convergent presentation on the two generators s and t. However, Bauer and Otto, [BO84], have found a ﬁnite + convergent presentation of the monoid B3 by adjunction of a new generator a standing for the product st:

Γ = s, t, a α : ta as, β : st a .

Indeed, this rewriting system can be completed by applying⇒ the Knuth-Bendix⇒ completion procedure, [KB70], into the following convergent presentation

KB(Γ) = s, t, a α : ta as, β : st a, γ : sas aa, δ : saa aat (2.6.7) with the following four critical branchings:⇒ ⇒ ⇒ ⇒ aaaβ βa aa γt aat γas aaas γaa aaaaey aaast (< K_ (< K_ )= au aaα *> EY sta γ sast sasas aata sasaa aaαt δ 6J sα "6 sas saβ "6 saa saγ !5 saaa δa saδ 4 saaat %9 aatat δat (2.6.8) + As a consequence, the word problem for B3 is solvable by the normal form algorithm. The result of Kapur and Narendran shows that the existence of a finite convergent presentation depends on the specific presentation of the monoid, in particular on the chosen generators. In their example, by adding new letters in the alphabet it is possible to obtain a finite convergent string rewriting system. However, is it always possible to obtain such Tietze equivalent system by adding a finite set of letters? Thus, to provide the awaited negative answer to the open question, one would have to exhibit a monoid with a decidable word problem but with no finite convergent presentation on any possible set of generators.

2.6.9. Question. Which condition a monoid need to satisfy to admit a presentation by a finite convergent rewriting system? Diekert solved the problem for the case of abelian groups. He derived a whole class of finite string rewriting systems presenting abelian groups with a decidable word problem which are not Tietze equivalent to finite convergent string rewriting systems on the same alphabet, [Die86]. Moreover, he constructed a necessary and sufficient conditions for the existence of convergent presentation for finitely generated abelian groups. However, the problem for general monoids was still open. At this point, new methods had to be introduced for a problem which seems to concern intrinsic properties of the presented monoid.

+ 2.6.10. Exercise. Compute a convergent presentation of the monoid B3 with two generating 1-cells.

2.6.11. Exercise, [KN85]. Consider the monoid B+ of positive braids on three strands and the Artin’s 3 presentation Σ = s, t γ : sts tst . + 1. Show that the word problem is decidable for B3 . ⇒ i+1 j+2 i+2 j+1 + 2. Show that for any i > 0 and any j > 0, the words s t st and tst s are equals in B3 .

33 2.6. Existence of ﬁnite convergent presentations

3. Denote by [w] the equivalence class modulo the relation γ containing the word w. Prove that for any n > 0 the two following equalities hold

n n−i i [t st] = { t sts | 0 6 i 6 n }. n j n−j [tst ] = { s tst | 0 6 j 6 n }.

+ 4. Show that there does not exist any ﬁnite convergent presentation of the monoid B3 with two generators s and t.

2.6.12. Example: plactic monoid. The structure of plactic monoids appeared in the combinatorial study of Young tableaux by Schensted [Sch61] and Knuth [Knu70]. The plactic monoid of rank n > 0, denoted by Pn, is generated by the set {1, . . . , n} and subject to the Knuth relations:

zxy = xzy for 1 6 x 6 y < z 6 n, yzx = yxz for 1 6 x < y 6 z 6 n.

For instance, the monoid P2 is generated by {1, 2} and submitted to the relations 211 = 121 and 221 = 212. These relations can be oriented with respect to the lexicographic order as follows

η1,1,2 : 211 121 ε1,2,2 : 221 212.

P In this way, the Knuth presentation of the monoid⇒ 2 is convergent⇒ with a unique critical branching:

2η1,1,2 . 2211 2121 . 1E

ε1,2,21

With respect to the lexicographic order, the Knuth presentation of the monoid P3 is not convergent, but it can be completed by adding 3 relations to get a convergent presentation with 27 critical branchings. For the monoid P4 we have 4 generators and 20 relations, and its completion is infinite. More generally, Kubat and Okninski´ showed in [KO14] that for rank n > 3, a finite convergent presentation of the monoid Pn cannot be obtained by completion of the Knuth presentation with the degree lexicographic order. Bokut, Chen, Chen and Li in [BCCL15], Cain, Gray and Malheiro in [CGM15], and Hage in [Hag15] for type C, constructed with independent methods a finite convergent presentation by adding column generators to the Knuth presentation.

34 CHAPTER 3

Coherent presentations and syzygies

Contents 3.1 Introduction ...... 35 3.2 Categorical preliminaries ...... 37 3.3 Coherent presentation of categories ...... 40 3.4 Finite derivation type ...... 42 3.5 Coherence from convergence ...... 44

The notion of coherent presentation extends those of presentation of a category by globular homotopy generators taking into account the relations amongst the relations. In this chapter we show how to compute a coherent presentation for a category using the completion procedure introduced in the previous chapter. The method follows a construction introduced by Squier in [SOK94] in his homotopical and homological study of finiteness conditions for finite convergence of finitely presented monoids. Many constructions presented in this chapter come from [GM18].

3.1. INTRODUCTION

3.1.1. Syzygies of Knuth’s relations. Recall from (2.6.12), that for n > 0, the plactic monoid Pn is generated by the set {1, . . . , n} and subject to the Knuth relations:

zxy = xzy for 1 6 x 6 y < z 6 n, yzx = yxz for 1 6 x < y 6 z 6 n. We consider the problem of ﬁnding all independent irreducible algebraic relations amongst these relations. Such a relation is called a 2-syzygy, and we aim to to give an algorithmic method that computes all

35 3.1. Introduction

2-syzygies of the presentation, and in particular a family of generators for these syzygies. For instance, the monoid P2 is generated by {1, 2} and submitted to the relations

η1,1,2 : 211 121, ε1,2,2 : 221 212.

There are two ways to prove the equality⇒2211 = 2121 in the monoid⇒ P2, either by applying the ﬁrst relation or the second relation. This two equalities are related by a syzygy:

2η1,1,2 . 2211 2121 Õ 1E ε1,2,21

We will prove that this syzygy generates all the syzygies of the above presentation. The proof is based on a categorical description of syzygies of such a presentation, and an extension of the Knuth-Bendix completion procedure given in (2.5.1), by keeping track of syzygies created when adding rules during the completion. The correctness of the procedure follows the coherent Squier theorem, [SOK94], which states that a convergent presentation of a monoid extended by the homotopy generators deﬁned by the conﬂuence diagrams induced by critical branchings forms a coherent convergent presentation.

3.1.2. Positive braid monoids. Let us illustrate the notion of syzygy on the presentation of the braid + monoid B3 studied in (2.6.6):

s, t α : sts tst .

One proves that there is no nontrivial syzygy amongst the⇒ relations induce by the rule α. Now consider + the braid monoid B4 on four strands with the following presentation:

h r, s, t | rsr = srs, sts = tst, rt = tr i.

The generators corresponds to the following generating braids on four strands:

r = , s = , t = . so that the relations read as follows:

= , = , = .

In that case, one proves [Del97, GGM15], that all the syzygies are generated by the following

36 3.2. Categorical preliminaries

Zamolodchikov relation:

%9 %9 %9 %9 0D Zn

Õ ;O

. %9 %9 %9 %9

3.2. CATEGORICALPRELIMINARIES

In this section, we recall the notion of 2-functor, (2, 1)-category, and 3-category used in this chapter.

3.2.1. Two-dimensional functors. In (2.1.13), we have introduced the notion of 2-category as a category enriched over the cartesian monoidal category Cat of categories. A (strict) 2-functor between 2-categories is a functor enriched in categories. Explicitly, given two 2-categories C and D.A 2-functor F : C D is a data made of

i) a map F0 : C0 D0, → ii) for every 0-cells x and y of C0, a functor → Fx,y : C(x, y) D(F0(x),F0(y)), such that the following diagrams commute in the category→ Cat, for every 0-cells x, y, z in C0 ?x,y,z C(x, y) × C(y, z) 0 / C(x, z)

Fx,y × Fy,z Fx,z D(F0(x),F0(y)) × D(F0(y),F0(z)) / D(F0(x),F0(z)) F0(x),F0(y),F0(z) ?0

| & C(x, x) / D(F0(x),F0(x)) Fx,x

37 3.2. Categorical preliminaries

where 1 denotes the terminal category:

Ô • with a single 0-cell and a single 1-cell, and the downward arrows map the single 0-cell on identities 1-cells.

If there is no possible confusion, we will write F(x) for F0(x) and F(u) (resp. F(α)) for Fx,y(u) (resp. Fx,y(α)), where u and α are 1-cells and 2-cells of C respectively. The 2-categories and their 2-functors form a category that we will denote by 2Cat.

3.2.2. (2, 1)-categories. A (small) groupoid, or (1, 0)-category, is a 1-category C in which all 1-cells −1 are isomorphisms, that is there is an inverse map (−) : C1 C1 such that for any 1-cell u in C1, the following conditions hold: − − uu = 1s0(u), u u =→1t0(u). A (2, 1)-category is a category enriched over the cartesian monoidal category Gpd of groupoids. That is, it is a 2-category C2, whose 2-cells are invertible for the 1-composition: for any 2-cell f : u v, there exists a 2-cell f−1 : v u, such that − − ⇒ f ?1 f = 1u, f ?1 f = 1v. ⇒ > 3.2.3. Free (2, 1)-category. Given a 2-polygraph Σ, the free (2, 1)-category over Σ is denoted by Σ2 and deﬁned as the free 2-category generated by Σ, and whose every 2-cell is invertible. Explicitly, its set > of 0-cells is Σ0 and, for all 0-cells x and y, the groupoid Σ2 (x, y) is given as the quotient > − ∗ Σ2 (x, y) = Σ q Σ ) (x, y) Inv(Σ2), where: i) the 2-polygraph Σ− is deﬁned from Σ by reversing its 2-cells, that is − Σ2 = { t1(α) s1(α) | α ∈ Σ2 },

ii) the cellular extension Inv(Σ2) contains the following two relations for every 2-cell α of Σ and all ∗ ⇒ possible 1-cells u and v of Σ1 such that s(u) = x and t(v) = y: − − uαv ?1 uα v ≡ 1us(α)v and uα v ?1 uαv ≡ 1ut(α)v.

> ∗ By deﬁnition of the (2, 1)-category Σ2 , for all 1-cells u and v of Σ1, we have u = v in the quotient > category Σ if, and only if, there exists a 2-cell f : u v in the (2, 1)-category Σ2 . 3.2.4. Lemma. Let C be a category and let Σ and Υ be two 2-polygraphs that present C. There exist two 2-functors ⇒ > > > > F : Σ2 Υ2 and G : Υ2 Σ2 and, there exist two families of 2-cells → → σu : GF(u) u ∗ and τv : FG(v) v ∗ u∈Σ1 v∈Υ1 in Σ> and Υ> respectively, such that the following conditions are satisﬁed: 2 2 ⇒ ⇒

38 3.2.5. 3-categories

i) the 2-functors F and G induce the identity through the canonical projections onto C, that is the two following diagrams commute

π π Σ> Σ / / C Σ> Σ / / C 2 2O O F IdC G IdC > / / > / / Υ2 C Υ2 C πΥ πΥ

ii) the 2-cells σu and τv are functorial in u and v, that is

0 0 σuu = σuσu , σ1x = 11x ,

for any 1-cells u and u0 and 0-cell x and

0 0 τvv = τvτv , τ1y = 11y ,

for any 1-cells v and v0 and 0-cell y.

Proof. We prove the existence of the functor F. The proof of the existence of the functor G is similar. For a 0-cell x, we set F(x) = x. If a : x y is a 1-cell of Σ, we choose, in an arbitrary way, a 1-cell ∗ ∗ F(a): x y in Υ1 such that πΥF(a) = πΣ(a). Then, we extend F to every 1-cell of Σ1 by functoriality. 0 0 Let α : u u be a 2-cell of Σ2. Since Σ→is a presentation of the category C, we have πΣ(u) = πΣ(u ), 0 so that πΥ→F(u) = πΥF(u ) holds. Using the fact that Υ is a presentation of the category C, we arbitrarily 0 > > choose a 2⇒-cell F(α): F(u) F(u ) in the (2, 1)-category Υ2 . Then, we extend F to every 2-cell of Σ2 by functoriality. Now, let us deﬁne σ, the⇒ case of τ being symmetric. Let a be a 1-cell of Σ. By construction of F and G, we have: πΣGF(a) = πΥF(a) = πΣ(a). > Since Σ is a presentation of C, there exists a 2-cell σa : GF(a) a in Σ2 . We extend σ to every 1-cell u > of Σ2 by functoriality. ⇒ 3.2.5. 3-categories. The notion of 3-category is deﬁned as the one of 2-category but by replacement of the hom-categories and the composition functors by hom-2-categories and composition 2-functors. A (strict) 3-category is a category enriched in the category 2Cat of 2-categories. In particular, in a 3-category, the 3-cells can be composed in three different ways:

i) by ?0, along their 0-dimensional boundary: u v uv

A B AB x 0 y g 0 z 7− x fg 0 0 z f %9 f A %9 g A %9 f g > Õ Õ Õ Õ Õ Õ → u0 v0 u0v0

39 3.3. Coherent presentation of categories

ii) by ?1, along their 1-dimensional boundary: u u A f %9 f0 A ?1 B x Õ v Õ / y 7− x f ? g 0 0 y B 1 %9 f ?1 g @ g %9 g0 Õ B Õ → Õ Õ w w iii) by ?2, along their 2-dimensional boundary: u u

A B A ?2 B g x f %9 %9 h > y 7− x f %9 h > y Õ Õ Õ Õ Õ v → v

The compositions in a 3-category satisfy the exchange relation, for every 0 6 i < j 6 2:

0 0 0 0 (A ?i B) ?j (A ?i B ) = (A ?j A ) ?i (B ?j B ).

3.2.6. (3, 1)-categories. A (3, 1)-category is a 3-category whose 2-cells are invertible for the composition ?1 and whose 3-cells are invertible for the composition ?2.

3.2.7. Exercise. Show that in a (3, 1)-category, all the 3-cells are invertible for the composition ?1.

3.3. COHERENT PRESENTATION OF CATEGORIES

3.3.1. Cellular extension of 2-categories. Let C be a 2-category. A 2-sphere of C is a pair (f, g) of parallel 2-cells of C, that is such that s1(f) = s1(g) and t1(f) = t1(g):

6J ' g

A cellular extension of the 2-category C is a set Γ equipped with a map from Γ to the set of 2-spheres of C. It is equivalent to the data of a set Γ with two maps

s o 2 C2 o Γ t2

40 3.3.2. Acyclicity

satisfying the globular relations s1s2 = s1t2 and t1s2 = t1t2. A congruence on the 2-category C is an equivalence relation ≡ on the parallel 2-cells of C such that, for every cells

h Õ 0 w / # w / x y f g ; Ez t Õ Õ k Õ of C, if f ≡ g, then 0 0 w ?0 (h ?1 f ?1 k) ?0 w ≡ w ?0 (h ?1 g ?1 k) ?0 w .

If Γ is a cellular extension of C, the congruence generated by Γ is denoted by ≡Γ and deﬁned as the smallest congruence such that, if Γ contains a 3-cell γ : f V g, then f ≡Γ g. The quotient 2-category of a 2-category C by a congruence relation ≡ is the 2-category, denoted by C/ ≡, whose 0-cells and 1-cells are those of C and the 2-cells are the equivalence classes of 2-cells of C modulo the congruence ≡.

3.3.2. Acyclicity. A cellular extension Γ of a 2-category C is called acyclic if for every parallel 2-cells f and g of C, we have f ≡Γ g, that is, the equality f = g holds in the quotient 2-category C/ ≡Γ . For instance, the set of 2-spheres of C forms an acyclic extension of C. In the literature, an acyclic extension of C is also called an homotopy basis of C.

3.3.3. (3, 1)-polygraphs. A (3, 1)-polygraph is a data (Σ0,Σ1,Σ2,Σ3) made of a 2-polygraph (Σ0,Σ1,Σ2) > and a cellular extension Σ3 of the free (2, 1)-category Σ2 over Σ2, as summarised in the following diagram: s s s o 0 ∗ o 1 > o 2 Σ0 o Σ1 o Σ2 o Γ3 t0 t1 t2

3.3.4. Coherent presentations. A coherent presentation of a 1-category C is a (3, 1)-polygraph (Σ0,Σ1,Σ2,Σ3) such that the 2-polygraph (Σ0,Σ1,Σ2) is a presentation of C and Σ3 is an acyclic cellular > extension of the (2, 1)-category Σ2 .

3.3.5. Free (3, 1)-categories. Given a (3, 1)-polygraph Σ, the free (3, 1)-category over Σ is denoted > by Σ3 and deﬁned as follows:

> i) its underlying 2-category is the free (2, 1)-category Σ2 ,

ii) its 3-cells are all the formal compositions by ?0, ?1 and ?2 of 3-cells of Σ3, of their inverses and of identities of 2-cells, up to associativity, identity, exchange and inverse relations.

> In particular, we get that Σ3 is an acyclic extension of Σ2 if, and only if, for every pair (f, g) of parallel > > 2-cells of Σ2 , there exists a 3-cell A : f V g in Σ3 .

41 3.4. Finite derivation type

3.4. FINITE DERIVATION TYPE

3.4.1. 2-polygraphs of finite derivation type. A 2-polygraph Σ is of finite derivation type, FDT for > short, if it is finite and if the free (2, 1)-category Σ2 admits a finite acyclic cellular extension. A category C is said to be of finite derivation type if it admits a finite coherent presentation. Let us prove now that this property does not depend on this presentation provide is finite. The proof is based on the following theorem, that allows transfers of acyclic cellular extensions of two (2, 1)-categories that present the same category.

3.4.2. Homotopy bases transfer theorem. Given a category C a category, we consider two presentations Σ and Υ of C. By Lemma 3.2.4, there exist 2-functors

> > > > F : Σ2 Υ2 and G : Υ2 Σ2

∗ > and for every 1-cell v of Υ2, there exists→ a 2-cell τv : FG(v) v in→Υ2 that satisfy the conditions given in Lemma 3.2.4. > Let deﬁne the cellular extension τΥ of the (2, 1)-category⇒Υ2 that contains one 3-cell

FG(v) FG(α) )= τv

FG(u) τα , v 1E Õ τu #7 u α for every 2-cell α : u v of Υ2. > Given a cellular extension Γ of the (2, 1)-category Σ2 , we denote by F(Γ) the cellular extension of > Σ2 that contains one 3⇒-cell F(f)

/ F(u) F(γ) F(v) Õ /C F(g) for every 3-cell γ : f V g of Γ. Using these notations, we can formulate the following result, called the acyclicity transfer theorem in [GM18].

> 3.4.3. Theorem. If Γ is an acyclic cellular extension of the (2, 1)-category Σ2 , then the cellular extension

∆ = F(Γ) t τΥ

> is an acyclic cellular extension of the (2, 1)-category Υ2 .

42 3.4.2. Homotopy bases transfer theorem

> > Proof. Let us deﬁne, for every 2-cell f : u v of Υ2 , a 3-cell τf of the free (3, 1)-category ∆3 with the following shape: FG(v) FG⇒(f) )= τv

FG(u) τf , v 1E Õ τu #7 u f

We extend the notation τα, where α is a 2-cell of Υ2 in a functorial way, according to the following formulas: − − − − τ1u = 1τu , τfg = τfτg, τf = FG(f) ?1 τf ?1 f , τf?1g = FG(f) ?1 τg ?2 τf ?1 g .

One checks that the 3-cells τf are well-deﬁned, i.e., that their deﬁnition is compatible with the relations on 2-cells, such as the exchange relation:

τfg?1hk = τ(f?1h)(g?1k). > > Now, let us consider parallel 2-cells f, g : u v of Υ2 . The 2-cells G(f) and G(g) are parallel in Σ2 > so that, since Γ is an acyclic cellular extension of Σ2 , there exists a 3-cell ⇒ G(f)

. G(u) A G(v) Õ 0D G(g)

> in Γ3 . An application of F to A gives the 3-cell FG(f)

. FG(u) F(A) FG(v) Õ 0D FG(g)

> which, by deﬁnition of the cellular extension ∆ and functoriality of F, is in ∆2 . Using the 3-cells F(A), > τf and τg, we get the following 3-cell from f to g in ∆3 :

f − − τu ?1 τf Õ FG(f) − τu * τv u %9 FG(u) F(A) FG(v) %9 ( v Õ 3G 6J FG(g) − τu ?1 τg Õ g

43 3.5. Coherence from convergence

This concludes the proof that ∆ = F(Γ) q τΥ is an acyclic cellular extension of the (2, 1)-category > Υ2 . We deduce from Theorem 3.4.3 that the ﬁnite derivation type property is Tietze invariant for ﬁnite 2- polygraphs:

3.4.4. Theorem ([SOK94, Theorem 4.3]). Let Σ and Υ be two Tietze-equivalent finite 2-polygraphs. Then Σ is of finite derivation type if and only if Υ is of finite derivation type.

The result of the following exercise is useful to prove that a presentation admits no ﬁnite acyclic cellular extensions.

3.4.5. Exercise. Let Σ be a 2-polygraph and let Γ be an acyclic cellular extension of the free (2, 1)- > > category Σ2 . Show that if Σ2 admits a ﬁnite acyclic cellular extension, then there exists a ﬁnite subset > of Γ that is an acyclic cellular extension of Σ2 .

3.5. COHERENCEFROMCONVERGENCE

3.5.1. Generating confluences. Squier’s completion procedure provides a way to extend a convergent presentation of a 1-category C into a coherent presentation of C. We fix a convergent 2-polygraph Σ.A > family of generating confluences of Σ is a cellular extension of the free (2, 1)-category Σ2 that contains exactly one 3-cell f '; v f0

, 0 u Af,g u Õ 3G g #7 w g0 for every critical branching (f, g) of Σ. Note that, if Σ is confluent, it always admits a family of generating confluences. However, such a family is not necessarily unique, since the 3-cell Af,g can be directed in the reverse way and, for a given branching (f, g), we can have several possible 2-cells f0 and g0 with the required shape. Later, we will define the notion of normalisation strategies that provide a deterministic way to construct a family of generating confluences.

3.5.2. Squier’s completion for convergent presentations. A Squier’s completion of a convergent 2- polygraph Σ is the (3, 1)-polygraph denoted by S(Σ) and defined by S(Σ) = (Σ, Γ), where Γ is a chosen family of generating confluences of Σ. The first proof of the following result is due to Squier, [SOK94], in the case where the category C is a monoid. We present the proof given in [GM18] in the language of polygraphs.

3.5.3. Theorem ([SOK94, Theorem 5.2]). For every convergent presentation Σ of a category C, Squier’s completion of Σ is coherent presentation of C.

Proof. We proceed in three steps.

44 3.5.4. Step 1

3.5.4. Step 1. We prove that, for every local branching (f, g): u (v, w) of Σ, there exist 2-cells 0 0 0 0 ∗ 0 0 > f : v u and g : w u in Σ2 and a 3-cell A : f ?1 f V g ?1 g in S(Σ)3 , as in the following diagram: ⇒ 0 ⇒ ⇒ f '; v f , u A u0 Õ 3G g #7 w g0 As we have seen in the study of conﬂuence of local branchings, in the case of an aspherical or Peiffer 0 0 0 0 branching, we can choose f and g such that f ?1 f = g ?1 g : an identity 3-cell is enough to link them. Moreover, if we have an overlapping branching (f, g) that is not critical, we have (f, g) = (uhv, ukv) 0 0 with (h, k) critical. We consider the 3-cell α : h ?1 h V k ?1 k of S(Σ) corresponding to the critical branching (h, k) and we conclude that the following 2-cells f0 and g0 and 3-cell A satisfy the required conditions: f0 = uh0v g0 = uk0v A = uαv.

∗ 3.5.5. Step 2. We prove that, for every parallel 2-cells f and g of Σ2 whose common target is a normal > form, there exists a 3-cell from f to g in S(Σ)3 . We proceed by noetherian induction on the common source u of f and g, using the termination of Σ. Let us assume that u is a normal form: then, by deﬁnition, > both 2-cells f and g must be equal to the identity of u, so that 11u : 1u V 1u is a 3-cell of S(Σ)3 from f to g. Now, let us ﬁx a 1-cell u with the following property: for any 1-cell v such that u rewrites into v ∗ > and for any parallel 2-cells f, g : v bv = ub of Σ2, there exists a 3-cell from f to g in S(Σ)3 . Let us consider parallel 2-cells f, g : u ub and let us prove the result by progressively constructing the > following composite 3-cell from f to ⇒g in S(Σ)3 : ⇒ f = u /C 1 f2 f f0 1 1 B ) 0 Õ -$ u A u h %9 u 5I 1E :Nb Õ g 0 C 1 g1 Õ / g2 v1 = g

Since u is not a normal form, we can decompose f = f1 ?1 f2 and g = g1 ?1 g2 so that f1 and g1 are 0 0 rewriting steps. They form a local branching (f1, g1) and we build the 2-cells f1 and g1, together with 0 ∗ the 3-cell A as in the ﬁrst part of the proof. Then, we consider a 2-cell h from u to ub in Σ2, that must exist by conﬂuence of Σ and since ub is a normal form. We apply the induction hypothesis to the parallel 0 0 2-cells f2 and f1 ?1 h in order to get B and, symmetrically, to the parallel 2-cells g1 ?1 h and g2 to get C.

45 3.5. Coherence from convergence

> > 3.5.6. Step 3. We prove that every 2-sphere of Σ2 is the boundary of a 3-cell of S(Σ)3 . First, let us ∗ consider a 2-cell f : u v in Σ2. Using the conﬂuence of Σ, we choose 2-cells

σ : u u and σ : v v = u ⇒ u b v b b ∗ in Σ2. By construction, the 2-cells f ?1 σv and σu are parallel and their common target ub is a normal ⇒ > ⇒ form. Thus by Step 2, there exists a 3-cell in S(Σ)3 from f ?1 σv to σu or, equivalently, a 3-cell σf from − > f to σu ?1 σv in S(Σ)3 , as in the following diagram:

u . v σf

> − − Moreover, the free (3, 1)-category S(Σ)3 contains a 3-cell σf− from f to σv?1σu , given as the following composite: u − σu +? b σv − − − f σ σv σu v u f v ! u u %9 Õ 0D %9 b %9 f

> > Now, let us consider a general 2-cell f : u v of Σ2 . By construction of the free (2, 1)-category Σ2 , the 2-cell f can be decomposed into a “zig-zag”, that is non-unique in general, ⇒ − − − f1 g1 f2 gn−1 fn gn u %9 v1 %9 u2 %9 (··· ) %9 un %9 vn %9 v

∗ > where each fi and gi is a 2-cell of Σ2. We deﬁne σf as the following composite 3-cell of S(Σ)3 , with − source f and target σu ?1 σv :

− − f1 g1 fn gn u %9 v1 %9 (··· ) %9 vn %9 v 2F 2F 2F :N σ − σ − − σ − σ − f1 σ σv g1 σ σu fn σ σv gn Õ v1 =1 Õ u2 n Õ vn = n Õ σu − 2 , , , σv ub ub (··· ) ub ub

> − > We proceed similarly for any other 2-cell g : u v of Σ2 , to get a 3-cell σg from g to σu ?1 σv in S(Σ)3 . − > Thus, the composite σf ?2 σg is a 3-cell of the free (3, 1)-category S(Σ)3 from f to g, concluding the proof. ⇒ Theorem 3.5.3 is extended to higher-dimensional polygraphs in [GM09, Proposition 4.3.4].

3.5.7. Theorem ([SOK94, Theorem 5.3]). If a monoid admits a ﬁnite convergent presentation, then it is of ﬁnite derivation type.

46 3.5.8. Example

+ 3.5.8. Example. Consider the convergent presentation KB(Γ) of the braid monoid B3 given in (2.6.7). It has four critical branchings given in (2.6.8). We deduce an acyclic extension of the (2, 1)-category KB(Γ)>, with the following 3-cells:

aaaβ βa aa γt aat γas aaas γaa aaaaey aaast (< K_ (< K_ )= au aaα *> EY sta A γ sast sasas C aata sasaa D aaαt B δ 6J Õ Õ Õ Õ sα "6 sas saβ "6 saa saγ !5 saaa δa saδ 4 saaat %9 aatat δat (3.5.9)

3.5.10. Example [LP91]. Consider the following 2-polygraph:

Σ = a, b, c, d α : ab a, β : da ac .

The 2-polygraph Σ is not convergent and can be completed⇒ into the⇒ following infinite but convergent polygraph n n KB(Σ) = a, b, c, d αn : ac b ac , n ∈ N, β : da ac , with an infinity of confluent critical branchings: ⇒ ⇒ n n dαn )= dac βc , dacnb A acn+1 n 5I Õ n α βc b 3 acn+1b n+1

By Theorem 3.5.3, the 2-polygraph KB(Σ) can be extended into a coherent presentation of the monoid Σ presented by Σ with inﬁnitely many 3-cells An, for n in N. Now, consider the following 2-polygraph

Γ = a, b, c, d α : ab a, γ : ac da .

It presents the monoid Σ and it is convergent with no critical⇒ branching.⇒ It follows that it forms a coherent presentation of the monoid Σ with no 3-cell.

3.5.11. Exercise. The standard presentation of a category C is the 2-polygraph Std2(C) deﬁned as follows. The 0-cells and 1-cells of Std2(C) are the ones of C, with ub denoting a 1-cell u of C when seen as a 1-cell of Std2(C). The 2-polygraph Std2(C) contains a 2-cell

7 y ub bv γ u,v x Õ 4 z uvc

47 3.5. Coherence from convergence for all 1-cells u : x y and v : y z of C, and a 2-cell

→ → 1x x ι x Õ x ?

b1x for every 0-cell x of C. Extend this 2-polygraph into a coherent presentation of the category C.

3.5.12. Exercise. Let us consider the monoid M presented by the 2-polygraph

Σ = x, y α : xyx yy .

1. Prove that Σ terminates. ⇒ 2. Complete Σ into a coherent presentation of the monoid M.

3.5.13. Squier’s example. In [Squ87], Squier deﬁnes, for every k > 1, the monoid Sk presented by the 2-polygraph

a, b, t, x1, . . . , xk, y1, . . . , yk (αn)n∈N, (βi)16i6k, (γi)16i6k, (δi)16i6k, (εi)16i6k with αn βi γi δi εi n at b %9 1, xia %9 atxi, xit %9 txi, xib %9 bxi, xiyi %9 1.

In [SOK94], Squier proves the following ﬁniteness properties for the monoid S1. With similar arguments, the result extends to every monoid Sk, for k > 1.

3.5.14. Theorem ([SOK94, Theorem 6.7, Corollary 6.8]). For every k > 1, the monoid Sk satisﬁes the following properties:

i) it is ﬁnitely presented,

ii) it has a decidable word problem, iii) it is not of ﬁnite derivation type, iv) it admits no ﬁnite convergent presentation.

This result shows in particular, that the property of being decidable is not sufficient for finitely presented monoids to have a finite convergent presentation or to have finite derivation type. Let us prove the result in the case of the monoid S1, with the following infinite presentation:

Sq1 Σ = a, b, t, x, y (αn)n∈N, β, γ, δ, ε

48 3.5.15. Exercise whose rules are deﬁned by

αn β γ δ ε atnb %9 1, xa %9 atx, xt %9 tx, xb %9 bx, xy %9 1.

n n Sq1 ∗ We will denote by γn : xt t x the 2-cell of (Σ )2 deﬁned by induction on n as follows:

γ = 1 γ = γtn ? tγ . ⇒ 0 x and n+1 1 n

b n+1 Sq1 ∗ For every n, we write fn : xat at bx the 2-cell of (Σ )2 deﬁned as the following composite:

n n+1 βt b atγnb at δ xatnb ⇒ %9 atxtnb %9 atn+1xb %9 atn+1bx.

3.5.15. Exercise.

1. Show that the monoid S1 admits the following ﬁnite presentation:

a, b, t, x, y α0, β, γ, δ, ε .

2. Show that the monoid S1 has a decidable word problem.

3.5.16. Exercise, [GM18].

Sq Sq 1. Show that the 2-polygraph Σ 1 is convergent and Squier’s completion of Σ 1 contains a 3-cell An for every natural number n with the following shape:

atn+1bx +? fn αn+1x An n Õ ) xat b (< x xαn

2. Show that the monoid S1 is not of ﬁnite derivation type.

3.5.17. Exercise, [LP91, Laf95]. Consider the monoid M presented by the following 2-polygraph:

0 0 a, b, c, d, d α0 : ab a, β : da ac, γ : d a ac .

M Show that the monoid admits a finite presentation,⇒ it has⇒ a decidable word⇒ problem, yet it is not of finite derivation type and, as a consequence, it does not admit a finite convergent presentation.

49 3.5. Coherence from convergence

50 CHAPTER 4

Two-dimensional homological syzygies

Contents 4.1 Preliminaries on modules ...... 51 4.2 Monoids of ﬁnite homological type ...... 55 4.3 Squier’s homological theorem ...... 61 4.4 Homology of monoids with integral coefﬁcients ...... 63 4.5 Historical notes ...... 65

In this chapter we present the result obtained by Squier relating the ﬁnite-convergence of a string rewriting system with the homotopical type left-FP3, [Squ87]. The constructions developed in this chapter come from [GM18].

4.1. PRELIMINARIESONMODULES

In this section, we ﬁx a ring R. We will say “R-module” of “module” for “left R-module”. All the notions presented are deﬁned in the same manner for right R-modules since every right R-module is a left Rop-module, where Rop is the opposite ring. We will say “homomorphism” for a homomorphism of left R-modules. We refer the reader to [Lan02] or to [Rot09] for a deeper presentation and the proofs of the results given in this preliminary part on modules.

4.1.1. Exact sequences. Two homomorphisms of modules f g M0 − M − M00

→ →

51 4.1. Preliminaries on modules are exact at M if Im f = ker g. A sequence of homomorphisms

dn+1 dn · · · − Mn+1 − Mn − Mn−1 − ··· is exact if each adjacent pair of homomorphisms is exact. → → → → f f 4.1.2. Examples. If 0 M M0 is exact, then the map f is injective. If M M0 0 is exact, then f the map f is surjective. If the sequence 0 M M0 0 is exact, then the map f is an isomorphism. f g If the sequence M0 M→ M→00 is exact with f surjective and g injective, then→M = →0. → → →

4.1.3. Free modules.→ A R→-module M is free if it is a direct sum of copies of R. If M = i∈I Rxi, with R ' Rxi, the the set {xi | i ∈ I} is called a basis of M. It follows that each element x in M has a unique decomposition ` x = λixi, i∈I X where λi ∈ R and almost all λi are zero.

4.1.4. Proposition. Let X = {xi | i ∈ I} be a basis of a free module M. For any module N and any map f : X − N, there is a unique map fe : M − N extending f, i.e., such that the following diagram commutes: → → MO fe

~ ? NXo f 4.1.5. Proposition. Let X be a set. There exists a free R-module having X as a basis. 4.1.6. Proposition. Every R-module is a quotient of a free R-module. Proposition 4.1.6 says that any R-module M may be described by generators and relations in the following way. Given a free R-module F with basis X and given f : F − M be a surjective homomorphism of R-modules, we say that X is a set of generators of M and the kernel ker f is called its submodule of relations. →

4.1.7. Finitely generated modules. A R-module M is finitely generated if there is a finite subset {x1, x2, ..., xn} of M such that for all x in M, there exist r1, r2, ..., rn in R with x = r1x1+r2x2+...+rnxn. Then the set {x1, x2, ..., xn} is referred to as a generating set for M. The finite generators need not be a basis, since they need not be linearly independent over R.A R-module M is finitely generated if and only if there is a surjective homomorphism: Rn − M for some n. That is, M is a quotient of a free module of finite rank. →

52 4.1.9. Projective modules

4.1.8. Proposition. Let F, M and N be left R-modules. If F is free, ε : M − N is a surjective homomorphism and f : F − N is any homomorphism, then there exists a homomorphism fe: F − M such that following diagram commutes → → → F f e f ~ / / M ε N 0

As a consequence of Proposition 4.1.8, for any free R-module, the functor HomR(F, −) is exact.

4.1.9. Projective modules. A projective module is a module which behaves as the free module F in Proposition 4.1.8. More explicitly, a R-module P is projective if whenever ε : M N is a surjective homomorphism and f : P − N is any homomorphism, there exists a homomorphism fe : P − M making the following diagram commutative: → → → P f e f ~ / / M ε N 0

In particular, any free module is projective. The following result gives several ways to characterise projective modules.

4.1.10. Proposition. The following conditions are equivalent for a R-module P:

i) P is projective,

ii) if f : M P is a surjective homomorphism, then there exists h : P − M such that fh = IdP, iii) if f : M P is a surjective homomorphism, then M ' P ⊕ ker f, → → 0 00 iv) the functor HomR(P, −) is exact, that is for any exact sequence 0 M M M 0, the → 0 00 induced sequence 0 HomR(P, M ) HomR(P, M) HomR(P, M ) 0 is exact, → → → → v) P is a summand of a free module, that is there exists a free R-module F such that F ' P ⊕ Q, for → → → → some R-module Q1.

4.1.11. Proposition (Schanuel’s Lemma). Given exact sequences of R-modules

0 − K1 − P1 − M − 0,

0 − K − P − M − 0, → 2 → 2 → → where P1 and P2 are projective. Then K1 ⊕ P2 ' K2 ⊕ P1. → → → → 1by the equivalence, the R-module Q is necessarily projective.

53 4.1. Preliminaries on modules

4.1.12. Exercise. Prove Proposition 4.1.11.

4.1.13. Proposition (Generalised Schanuel’s Lemma). Given exact sequences of R-modules

0 − K − Pk − Pk−1 − · · · − P1 − P0 − M − 0,

0 − →L − →Qk − →Qk−1 −→ · · · −→ Q1 −→ Q0 −→ M −→ 0, where all the P and Q are projective. Let i → i → → → → → → →

Podd = ⊕ Pi,Peven = ⊕ Pi, i odd i even and

Qodd = ⊕ Qi,Qeven = ⊕ Qi. i odd i even Then the following properties hold

i) If k is even, then K ⊕ Qeven ⊕ Podd ' L ⊕ Qodd ⊕ Peven,

ii) If k is odd then K ⊕ Qodd ⊕ Peven ' L ⊕ Qeven ⊕ Podd.

Let us mention a consequence of the Proposition 4.1.13.

4.1.14. Corollary. Given exact sequences of R-modules

0 − K − Pk − Pk−1 − · · · − P1 − P0 − M − 0,

0 − →L − →Qk − →Qk−1 −→ · · · −→ Q1 −→ Q0 −→ M −→ 0, where all the P and Q are finitely generated and projective, then the R-module K is finitely generated if i → i → → → → → → → and only if L is finitely generated.

4.1.15. Exercise. Prove Proposition 4.1.13.

4.1.16. Chain’s complex. A (chain) complex of R-modules is a sequence (Mn)n∈N of R-modules, together with a sequence (dn)n∈N of homomorphisms

dn d2 d1 ··· / Mn / Mn−1 / ··· / M2 / M1 / M0 such that we have the inclusion

Im dn+1 ⊆ ker dn for all n, or equivalently, the relation dndn+1 = 0 holds for all n. The map dn are called boundary maps.

54 4.1.17. Resolutions

4.1.17. Resolutions. A resolution of a R-module M is an exact sequence of R-modules

dn d2 d1 ε ··· / Mn / Mn−1 / ··· / M2 / M1 / M0 / M / 0 From the deﬁnition, the homomorphism ε is surjective and

Im d1 = ker ε, and Im dn+1 = ker dn, for all n.

Such a resolution is called projective (resp. free) if all the modules Mn are projective (resp. free). Given a natural number n, a partial resolution of length n of M is deﬁned in a similar way but with a bounded sequence (Mk)06k6n of R-modules:

dn d2 d1 ε Mn / Mn−1 / ··· / M2 / M1 / M0 / M / 0 4.1.18. Proposition. Every R-module M has a free resolution.

4.1.19. Exercise. Prove Proposition 4.1.18.

4.1.20. Contracting homotopies. Given a complex of R-modules

dn+1 dn d1 ε ··· / Mn+1 / Mn / Mn−1 / ··· / M1 / M0 / M / 0 (4.1.21) a method to prove that such a complex is a resolution of M is to construct a contracting homotopy, that is a sequence of homomorphisms of Z-modules

in+1 in i1 i0 ··· o Mn+1 o Mn o Mn−1 o ··· o M1 o M0 o M satisfying the following equalities

εi0 = IdM,

d1ι1 + ι0ε = IdM0 ,

dn+1in+1 + indn = IdMn , for all n ∈ N. Indeed, in that case, the ﬁrst equality proves that the homomorphism ε is surjective. Moreover, for every natural number n and every x in ker dn, the equality dn+1in+1(x) = x holds, proving that x is in Im dn+1, so that ker dn ⊆ Im dn+1 holds. As a consequence, the complex (4.1.21) is a resolution of the R-module M.

4.2. MONOIDS OF FINITE HOMOLOGICAL TYPE

Let M be a monoid. We denote by ZM the ring generated by M, that is, the free abelian group over M, equipped with the canonical extension of the product of M: λuu λvv = λuλvuv = λuλvw, u∈M v∈M u,v∈M w∈M uv=w X X X X X with λu, λv in Z. The trivial ZM-module is the abelian group Z equipped with the trivial action un = n, for every u in M and n in Z.

55 4.2. Monoids of ﬁnite homological type

4.2.1. Homological type left-FPn. A monoid M is of homological type left-FPn, for a natural number n, if there exists a partial resolution of length n of the trivial ZM-module Z by projective, ﬁnitely generated ZM-modules:

dn dn−1 d2 d1 d0 Pn / Pn−1 / ··· / P1 / P0 / Z / 0.

A monoid M is of homological type left-FP if there exists a resolution of Z by projective, ﬁnitely generated ZM-modules. ∞ 4.2.2. Proposition. Let M be a monoid and let n be a natural number. The following assertions are equivalent:

i) The monoid M is of homological type left-FPn.

ii) There exists a free, ﬁnitely generated partial resolution of the trivial ZM-module Z of length n

Fn / Fn−1 / ··· / F0 / Z / 0. iii) For every 0 6 k < n and every projective, ﬁnitely generated partial resolution of the trivial ZM- module Z of length k

dk dk−1 d0 Pk / Pk−1 / ··· / P0 / Z / 0,

the ZM-module ker dk is ﬁnitely generated.

4.2.3. Exercise. Show Proposition 4.2.2 using Proposition 4.1.14.

4.2.4. Homological type FP0. The augmentation map of a monoid M is the ring homomorphism

ε : ZM Z deﬁned by → ε λuu = λu. u∈M u∈M X X The ring homomorphism ε is extended to a homomorphism of ZM-modules in the obvious way. If we consider the homomorphism of Z-modules i0 : Z ZM deﬁned by i0(1) = 1, we have εi0 = IdZ. Hence the homomorphism ε is surjective. It follows that → 4.2.5. Proposition. Every monoid M is of homological type FP0.

56 4.2.6. Remark

4.2.6. Remark. Every R-module admits a free resolution. In particular, given a monoid M, there exists a resolution

dn+1 dn d1 d0 ··· / Fn+1 / Fn / Fn−1 / ··· / F1 / F0 / Z / 0 of the trivial ZM-module Z by free ZM-modules. We can build such a resolution by setting F0 = ZM and d0 = ε. Let F1 be the free ZM-module generated by ker ε, and let d1 : F1 ZM be the canonical homomorphism induced by the homomorphism ker ε F0. Then, for any n > 2, Fn is the free ZM-module generated by ker dn−1, and the homomorphism dn : Fn Fn−1 is induced→ by the homomorphism ker dn−1 Fn−1. → Note that in this way, the obtained resolution is too big in general. In the rest→ of this section, we show how to construct a partial→ resolution which is more “economic” in the sense that the free modules are generated by a reduced number of generators.

4.2.7. Normalisation strategies. Given a monoid M, we consider a presentation of M by a 2-polygraph ∗ Σ with a single 0-cell • . Let π : Σ1 M be the canonical projection. We will write u instead of π(u). We consider a section ∗ → M Σ1 ∗ of π, i.e., we choose, for every 1-cell u of M, a 1-cell ub of Σ1 such that π(ub) = u. In general, we cannot → ∗ assume that the chosen section is functorial, that is uvc = ubbv holds in Σ1. However, we will assume ∗ that b1• = 1• holds. Given a 1-cell u of Σ1, we simply write ub for ub. Such a section being ﬁxed, a normalisation strategy for Σ is a map

∗ > σ : Σ1 Σ2

1 u Σ∗ 2 that sends every -cell of 1 to a -cell →

σu : u ub of the free (2, 1)-category Σ>, such that σ = 1 holds for every 1-cell u of Σ∗. 2 ub ub ⇒ 1

4.2.8. Left and right normalisation strategies. Let Σ be a 2-polygraph, with a chosen section. A normalisation strategy σ for Σ is a left one (resp. a right one) if it satisﬁes

σ = (σ ? v) ? σ , resp. σ = (u ? σ ) ? σ . (4.2.9) uv u 0 1 uvb uv 0 v 1 ubv That is u + v B • 4 • σu v u # { σv σ = u σuv resp. σ = σuv v . uv b Õ b uv Õ b b • 2 • • *2 • uvc uvc 4.2.10. Proposition. Any 2-polygraph admits a left (resp. right) normalisation strategy.

57 4.2. Monoids of ﬁnite homological type

Proof. Let Σ be a 2-polygraph with a chosen section. Prove that Σ admits a left normalisation strategy ∗ > σ : Σ1 Σ2 . The proof of the existence of a right normalisation strategies is similar. Let us arbitrarily choose a 2-cell σ : ua ua in Σ>, for every 1-cell u of Σ∗ and every 1-cell a uab b c 2 1 of Σ , such that ua 6= ua. Then, we extend σ into a left normalisation strategy for Σ by setting σ = 1 , 1 → c b ub ub ∗ for any u in Σ1, and for u 6= ub by setting ⇒ σ = σ a ? σ u v 1 bva ∗ if u = va with v in Σ1 and a in Σ1: v + B • σv a # v σva b Õ b • 2 • vac

The relations σ1• = 11• and (4.2.9) are immediate consequences of the deﬁnition of the map σ.

4.2.11. Leftmost and rightmost normalisation strategies. If Σ is a reduced 2-polygraph, then, for ∗ every 1-cell u of Σ1, the set of rewriting steps with source u can be ordered from left to right: for two rewriting steps f = vαv0 and g = wβw0 with source u, we have f ≺ g if the length of v is strictly smaller than the length of w. If Σ is ﬁnite, then the order ≺ is total and the set of rewriting steps of source u is ﬁnite. Hence, this set contains a smallest element λu and a greatest element ρu, respectively called the leftmost and the rightmost rewriting steps on u. If, moreover, the 2-polygraph Σ terminates, the iteration of λ (resp. ρ) yields a normalisation strategy σ called the leftmost (resp. rightmost) normalisation strategy of Σ:

σu = λu ?1 σt(λu) (resp. σu = ρu ?1 σt(ρu)). The leftmost and rightmost normalisation strategies give a way to make constructive some of the results we present here. For example, when Σ is convergent they provide a deterministic choice of a conﬂuence diagram f '; v σv ) u u 7K b

g "6 w σw for every branching (f, g) of Σ.

4.2.12. Exercice. Prove (by noetherian induction) that the leftmost (resp. rightmost) normalisation strategy of Σ is a left (resp. right) normalisation strategy.

4.2.13. Presentations and partial resolutions of length 2. Let M be a monoid and let Σ be a presentation of M. Let us deﬁne a partial resolution of length 2 of Z by free ZM-modules

d2 d1 ε ZM[Σ2] / ZM[Σ1] / ZM / Z / 0.

58 4.2.13. Presentations and partial resolutions of length 2

The ZM-modules ZM[Σ1] and ZM[Σ2] are the free ZM-modules over Σ1 and Σ2, respectively: they contain the formal sums of elements denoted by u[x], where u is an element of M and x is a 1-cell of Σ or a 2-cell of Σ. Let us note that ZM is isomorphic to the free ZM-module over the singleton Σ0. The map ε is the augmentation map deﬁned in (4.2.4) and the boundary maps are deﬁned, on generators, by

d1([x]) = x − 1 d2([α]) = [s1(α)] − [t1(α)].

The map d2 is called the Reidemester-Fox Jacobian of Σ. In the deﬁnition of d2, the bracket [·] is ∗ extended to the 1-cells of Σ1 thanks to the relation [1] = 0 and [uv] = [u] + u[v], (4.2.14) for all 1-cells u and v of Σ1. ∗ 4.2.15. Lemma. For any u in Σ1, we have d1(u) = u − 1.

Proof. We prove the relation by induction on the length of u. For the unit, we have d1[1] = d1(0) = 0 and 1 − 1 = 0. Then, for a composite 1-cell uv such that the result holds for both u and v, we get

d1[uv] = d1[u] + ud1[v] = u − 1 + uv − u = uv − 1.

4.2.16. Proposition. Let M be a monoid and let Σ be a presentation of M. The sequence of ZM-modules

d2 d1 ε ZM[Σ2] / ZM[Σ1] / ZM / Z / 0 (4.2.17) is a partial free resolution of length 2 of Z. Proof. We ﬁrst note that the sequence is a chain complex. Indeed, the augmentation map is surjective by deﬁnition. Moreover, we have

εd1[x] = ε(x) − ε(1) = 1 − 1 = 0, for every 1-cell x of Σ1. The relation d1d2 = 0 is consequence of Lemma 4.2.15. Indeed, we have

d1d2[α] = d1[s(α)] − d1[t(α)] = s(α) − t(α) = 0, for every 2-cell α of Σ2, where the last equality comes from s(α) = t(α), that holds since Σ is a presentation of the monoid M. The rest of the proof consists in deﬁning contracting homotopies i0, i1, i2:

d d 2 / 1 / ε / ZM[Σ2] o ZM[Σ1] o ZM o Z i2 i1 i0

∗ We choose a representative ub in Σ1 for every element u of M, with b1x = 1x for every 0-cell x of Σ, and we ﬁx a normalisation strategy σ for Σ. Then we deﬁne the homomorphisms of Z-modules i0, i1 and i2 by setting i (1) = 1, i (u) = [u], i (u[x]) = [σ ], (4.2.18) 0 1 b 2 uxb

59 4.2. Monoids of ﬁnite homological type

for any u in M and x in Σ1. In (4.2.18), the element [σ ] is deﬁned using an extension of the bracket notation [·] on 2-cells of Σ uxb 2 into a map > [·]: Σ2 ZM[Σ2] thanks to the relations → [1u] = 0, [ufv] = u[f] and [f ?1 g] = [f] + [g],

> for all 1-cells u and v and 2-cells f and g of Σ2 such that the composites ufv and f ?1 g are deﬁned. First, we have εi0 = IdZ. Next, for every u in M, we have i0ε(u) = 1 and

d1i1(u) = d1[ub] = u − 1.

Thus d1i1 + i0ε = IdZM. Finally, we have, on the one hand,

i1d1(u[x]) = i1(ux − u) = [uxc] − [ub] and, on the other hand,

d i (u[x]) = d [σ ] = [ux] − [ux] = u[x] + [u] − [ux]. 2 2 2 uxb b c b c > For this equality, we check that d2[f] = [s(f)] − [t(f)] holds for every 2-cell f of Σ2 by induction on the f d i + i d = size of . Hence we have 2 2 1 1 IdZM[Σ1], thus concluding the proof. From Proposition 4.2.16, we deduce the following result:

4.2.19. Theorem. The following properties hold.

i) Every monoid is of homological type left-FP0.

ii) Every ﬁnitely generated monoid is of homological type left-FP1. iii) Every ﬁnitely presented monoid is of homological type left-FP2.

4.2.20. Examples. Let us consider the monoid M presented by the 2-polygraph

n+1 n Σ = a, c, t αn : at ct , n ∈ N . The monoid M is ﬁnitely generated and, thus, it is of homological type left-FP . However, for every ⇒ 1 natural number n, we have

n+2 n+1 d2[αn+1] = [at ] − [ct ], = [atn+1] + atn+1[t] − [ctn] − ctn[t], n+1 n = d2[αn] + (at − ct )[t].

n+1 n The equality at = ct holds in M by deﬁnition, yielding d2[αn+1] = d2[αn]. As a consequence, the ZM-module ker d2 is generated by the elements [αn] − [α0]. Since the ZM-module ker d1 is equal

60 4.3. Squier’s homological theorem

to Im d2, hence isomorphic to ZM[Σ2]/ ker d2, it follows that ker d1 is generated by [α0] only, so that, by Lemma 4.2.2, the monoid M is of homological type left-FP2. This can also be obtained by simply observing that M admits the ﬁnite presentation ha, c, t | α0i. Now, let us consider the monoid M presented by the 2-polygraph

n Σ = a, b, t αn : at b 1 , n ∈ N .

The monoid M is of homological type left-FP1, but not left-FP⇒ 2. This is proved by showing that ker d1 is not ﬁnitely generated as a ZM-module, which is tedious by direct computation in this case. Another way to conclude is to extend the partial resolution of Proposition 4.2.16 by one dimension: it will then be sufﬁcient to compute Im d3, which is trivial in this case because Σ has no critical branching, so that ker d2 = 0 and, as a consequence, ker d1 is isomorphic to ZM[Σ2]. Convergent presentations provide a method to obtain such a length-three partial resolution.

4.3. SQUIER’SHOMOLOGICALTHEOREM

4.3.1. Coherent presentations and partial resolutions of length 3. Let M be a monoid and let Σ be a coherent presentation of M. Let us extend the partial resolution (4.2.17) into the resolution of length 3

d3 d2 d1 ε ZM[Σ3] / ZM[Σ2] / ZM[Σ1] / ZM / Z / 0, where the ZM-module ZM[Σ3] is the free ZM-module over Σ3, formed by the linear combination of elements u[γ], with u in M and γ a 3-cell of Σ3. The boundary map d3 is deﬁned, for every 3-cell γ of Σ3, by d3[γ] = [s2(γ)] − [t2(γ)].

The bracket notation [·] deﬁned on 3-cells of Σ3 can be extended into a map

> [·]: Σ3 ZM[Σ3] thanks to the relations →

[uAv] = u[A], [A ?1 B] = [A] + [B], [A ?2 B] = [A] + [B],

> for all 1-cells u and v and 3-cells A and B of Σ3 such that the composites are deﬁned. In particular, > the latter relation implies [1f] = 0 for every 2-cell f of Σ2 . We check, by induction on the size, that > d3[A] = [s2(A)] − [t2(A)] holds for every 3-cell A of Σ3 .

4.3.2. Proposition. Let M be a monoid and let Σ be a coherent presentation of M. The sequence of ZM-modules d3 d2 d1 ε ZM[Σ3] / ZM[Σ2] / ZM[Σ1] / ZM / Z / 0 is a partial free resolution of length 3 of Z.

61 4.3. Squier’s homological theorem

Proof. We proceed with the same notations as the ones of the proof of Proposition 4.2.16, with the extra hypothesis that σ is a left normalisation strategy for Σ. This implies that i (u[v]) = [σ ] holds for all u 2 uvb ∗ in M and v in Σ1, by induction on the length of v. We have d2d3 = 0 because s1s2 = s1t2 and t1s2 = t1t2. Then, we deﬁne the following homomorphism of Z-modules i3 ZM[Σ2] − ZM[Σ3] u[α] 7− [σ ] uαb ) → where σ is a 3-cell of Σ> with the following shape, with v = s(α) and w = t(α): uαb 3 →

uw -A b σ uαb uwb σ Õ uαb $ uvb (< uvc σ uvb

> Let us note that such a 3-cell necessarily exists in Σ3 because Σ3 is an acyclic extension of the free > (2, 1)-category Σ2 . Then we have, on the one hand,

i d (u[α]) = i (u[v] − u[w]) = [σ ] − [σ ] 2 2 2 uvb uwb and, on the other hand,

d i (u[α]) = [uα ? σ ] − [σ ], 3 3 b 1 uwb uvb = u[α] + [σ ] − [σ ]. uwb uvb

d i + i d = Hence 3 3 2 2 IdZM[Σ2], concluding the proof.

4.3.3. Remark. The proof of Proposition 4.3.2 uses the fact that Σ3 is an acyclic extension to produce, for every 2-cell α of Σ and every u in M, a 3-cell σ with the required shape. The hypothesis on Σ 2 uαb 3 > could thus be modiﬁed to only require the existence of such a 3-cell in Σ3 . It is proved in [GM12b] that > this implies that Σ3 is an acyclic extension of the free (2, 1)-category Σ2 . From Proposition 4.3.2, we deduce

4.3.4. Theorem ([CO94, Theorem 3.2], [Laf95, Theorem 3], [Pri95]). Let M be a ﬁnitely presented monoid. If M is of ﬁnite derivation type, then it is of homological type left-FP3.

By Theorem 3.5.7, this implies

4.3.5. Theorem ([Squ87, Theorem 4.1]). If a monoid admits a ﬁnite convergent presentation, then it is of homological type left-FP3.

62 4.3.6. Example

4.3.6. Example. Let us consider the monoid M with the convergent presentation

h a | µ : aa a i.

With the leftmost normalisation strategy σ, we get, writing⇒ the 2-cell µ as a string diagram :

σa = 1a σaa = σaaa = µa ?1 µ = .

The presentation has exactly one critical branching, whose corresponding generating conﬂuence can be written in the two equivalent ways

)= aa aaa %a or . Õ 8L %9 !5 aa

The ZM-module ker d2 is generated by d3 = − ] = + − − = a − .

4.4. HOMOLOGY OF MONOIDS WITH INTEGRAL COEFFICIENTS

4.4.1. Morphism of resolutions. Let M be a monoid. Consider two free resolutions of the trivial ZM- module Z by ZM-modules

dn+1 dn d1 ε F : ··· / Fn+1 / Fn / Fn−1 / ··· / F1 / F0 / Z / 0

d0 d0 d0 0 0 / 0 n+1 / 0 n / 0 / / 0 1 / 0 ε / / F : ··· Fn+1 Fn Fn−1 ··· F1 F0 Z 0

0 0 A homomorphism of resolutions f : F F is a family of homomorphisms f = (fn : Fn Fn)n∈N making the following diagrams commutative → → dn+1 dn d1 ε ··· / Fn+1 / Fn / Fn−1 / ··· / F1 / F0 / Z / 0

fn+1 fn fn−1 f1 f0 IdZ d0 d0 d0 0 / 0 n+1 / 0 n / 0 / / 0 1 / 0 ε / / ··· Fn+1 Fn Fn−1 ··· F1 F0 Z 0

63 4.4. Homology of monoids with integral coefﬁcients

4.4.2. Homotopy of resolutions. Given two homomorphisms of resolutions f, g : F F 0 given by

dn+1 dn d1 → ε ··· / Fn+1 / Fn / Fn−1 / ··· / F1 / F0 / Z / 0

g g g g g n+1 fn+1 n fn n−1 fn−1 1 f1 0 f0 IdZ d0 d0 d0 0 / 0 n+1 / 0 n / 0 / / 0 1 / 0 ε / / ··· Fn+1 Fn Fn−1 ··· F1 F0 Z 0

0 We say that f is homotopic to g if there exists a family of homomorphisms h = (hn : Fn Fn+1)n∈Z such that

0 → f0 − g0 = d1h0, 0 fn − gn = dn+1hn + hn−1dn, for all n > 1. It is easy to see that homotopy is an equivalence relation on the set of homomorphisms of resolutions from F to F 0.

4.4.3. Proposition. Between two free resolutions, there exists a homomorphism. Moreover, two such homomorphisms are homotopic.

4.4.4. Exercise. Prove Proposition 4.4.3.

4.4.5. Homology with integral coefﬁcients. Let M be a monoid. To a free resolution of the trivial ZM-module Z by left ZM-modules

dn+1 dn d1 ε ··· / Fn+1 / Fn / Fn−1 / ··· / F1 / F0 / Z / 0 we associate the following complex of Z-modules

d d d / en+1/ en / / / e1 / ··· Z ⊗ZM Fn+1 Z ⊗ZM Fn Z ⊗ZM Fn−1 ··· Z ⊗ZM F1 Z ⊗ZM F0 where den = Id ⊗dn. Note that the Z-module Z ⊗ZM Fn is obtained from Fn by trivialising the action of M, that is Fn quotiented by all relations ux = x for u ∈ M and x ∈ Fn. In particular, if Fn = ZM[X], then Z ⊗ZM Fn = Z[X] is the free Z-module on X. We obtain a chain complex, because dndn+1 = 0 induces that denden+1 = 0. We deﬁne the n-th homology group of M with integral coefﬁcient Z as the quotient Z-module:

Hn(M, Z) = ker(den)/Im (den+1), with the convention that d0 = 0. For any monoid M, we have H0(M, Z) ' Z.

4.4.6. Proposition. For n > 0, the group Hn(M, Z) does not depend on a particular choice of a free resolution, but only on the monoid M itself.

64 4.4.7. Exercise

4.4.7. Exercise. Prove the Proposition 4.4.6.

4.4.8. Proposition. If a monoid M is of homological type left-FPn for all n > 0, then the groups Hn(M, Z) are all ﬁnitely generated. In particular, we have the following consequence that gives a necessary condition for a monoid to have a ﬁnite convergent presentation.

4.4.9. Corollary. If a monoid admits a ﬁnite convergent presentation, then the group H3(M, Z) is ﬁnitely generated.

4.4.10. Exercise. Consider the monoid M presented by the following 2-polygraph:

n h a, b, c | αn : ac b 1, n ∈ N i. 1. Compute homology groups H (M, ) and H (M, ). 1 Z 2 ⇒Z 2. Show that M is a ﬁnitely generated monoid which cannot be ﬁnitely presented.

4.4.11. Exercise. Consider the monoid M presented by the following 2-polygraph:

h a, b, c, d | ab a, da ac i.

Compute the homology groups H (M, ), for n = 1, 2, 3. n Z ⇒ ⇒ 4.4.12. Exercise. Consider the monoid M presented by the following 2-polygraph:

h a, b, c, d, d0 | ab a, da ac, d0a ac i.

Compute the homology groups H (M, ), for n = 1, 2, 3. n Z ⇒ ⇒ ⇒

4.5. HISTORICALNOTES

4.5.1. Homological finiteness condition. Jantzen in [Jan82, Jan85] asked the following question: does every finitely presented monoid with a decidable word problem admit a finite convergent presentation? At the end of the eighties, using a homological argument Squier answered the Jantzen question negatively by showing that there are finitely presented monoids with a decidable word problem which do not have a finite convergent presentation, [SO87, Squ87]. He linked the existence of a finite convergent presentation for a finitely presented monoid to the homological type left-FP3 property, Theorem 4.3.5. He showed that a monoid needs to satisfy this invariant to have a finite convergent presentation. Giving examples, recalled in Example 3.5.13, of finitely presented monoids that have a decidable word problem and that do not have homological type left-FP3, he proved that there are finitely presented monoids with a decidable word problem that cannot be presented by a finite convergent string rewriting system. However, it still remains open to characterize the class of monoids with a decidable word problem and having a finite convergent presentation. Squier result leads to the following question: is the homological finiteness condition left-FP3 sufficient for a finitely presented monoid with a decidable word problem to admit a finite convergent presentation?

65 4.5. Historical notes

4.5.2. Homotopical finiteness condition. Squier answered this question negatively in an another article. In [SOK94], he related the existence of a finite convergent presentation to a new finiteness condition of finitely presented monoids, called finite derivation type, Definition 3.4.1. This property is a natural extension of the properties of being finitely generated and finitely presented. Squier defined the finite derivation type for a monoid as a finiteness property on a 2-dimensional combinatorial complex associated to a presentation of the monoid. Note that this complex was defined independently by Kilibarda, [Kil97], and Pride, [Pri95]. Squier proved that the finite derivation type property is an invariant property for finitely presented monoids, Theorem 3.4.4. As a consequence, the property finite derivation type can be defined for monoids independently of a considered presentation: a monoid is of finite derivation type if its finite presentations are of finite derivation type. The proof given by Squier is based on Tietze transformations. Finally, Squier proved that, if a monoid admits a finite convergent presentation, then it is of finite derivation type, Theorem 3.5.7. This result corresponds to a “homotopical” version of New- man’s Lemma 5.5.12 for string rewriting systems. Squier used this result to give another proof that there exist finitely presented monoids with a decidable word problem that do not admit a finite convergent presentation. Moreover, he showed that the homological finiteness condition left-FP3 is not sufficient for a finitely presented monoid with a decidable word problem to admit a finite convergent presentation. Indeed, he showed that the finitely presented monoid S1 given in Example 3.5.13 has a decidable word problem and is of homological type left-FP3, but it is not of finite derivation type, and, thus, it does not admit a finite convergent presentation. The article [SOK94] concludes with the following question: for finitely presented monoids does the property of having finite derivation type implies the existence of a finite convergent presentation? The answer is negative, indeed there exist finitely presented groups of homological type left-FP3 that have undecidable word problems, [Mil92]. Since for finitely presented groups the property of having finite derivation type is equivalent to the homological type left-FP3, [CO96], it follows that a finitely presented group can have an undecidable word problem even if it has finite derivation type. Hence in general the finite derivation property is not sufficient for the existence of a finite convergent presentation.

4.5.3. Extensions of Squier’s finiteness conditions. By his results, Squier has opened a homological direction and a homotopical one, in the quest for a complete characterisation of the existence of finite convergent presentations of monoids. In the homological direction, it has been shown that a finitely presented monoid admitting a finite convergent presentation satisfies the more restrictive condition homological type left-FP , Definition 4.2.1. Further proofs of the following result can be found in the literature. ∞ 4.5.4. Theorem ([Ani86, Kob90, Gro90, Bro92]). If a monoid admits a finite convergent presentation, then it is of homological type left-FP .

The proofs are based on distinct ways∞ to describe the n-fold critical branchings of a convergent rewriting system. Note that the converse implication of this result is false in general. By this fact, there were numerous finiteness conditions introduced with the goal to have a sufficient condition for the finite- convergence, [WP00, KO01, KO02, KO03, PO04, MPP05, GM13]. However, all these conditions were necessarily but not sufficient. The characterization of the class of finitely presented monoids having a presentation by a finite convergent rewriting system is still an open problem. Beyond this problem, the methods initiated by Squier have opened the way to homotopical and homological analysis of rewriting systems. Moreover, it was shown in [Ani86, Kob90, Gro90, Bro92,

66 4.5.5. Question ([LM09, LMW10])

Mal03, GHM19] that this methods highlight the way to compute “effectively” free resolutions for groups, monoids, associative algebras or small categories using rewriting. Finally, the question of putting all this work in a higher-categorical framework was posed by Lafont and Métayer, [Laf95, Mét03, LM09]. In particular, is it possible to describe in the higher-categorical framework the constructions developed in [Ani86, Kob90, Gro90, Bro92]:

4.5.5. Question ([LM09, LMW10]). Is it true that a monoid presented by a finite convergent rewriting system always has a finite cofibrant approximation in the folk model structure on -categories? We will see that in fact the higher-dimensional strict categories constitute a natural setting for the analysis of rewriting systems. ∞

67 4.5. Historical notes

68 CHAPTER 5

Linear rewriting

Contents 5.1 Linear 2-polygraphs ...... 70 5.2 Linear rewriting steps ...... 77 5.3 Termination for linear 2-polygraphs ...... 79 5.4 Monomial orders ...... 80 5.5 Conﬂuence and convergence ...... 82 5.6 The Critical Branchings Theorem ...... 85 5.7 Coherent presentations of algebras ...... 90

We must be careful when we rewrite in a linear structure deﬁned over a ﬁeld. For example, consider a rewriting system over a ring or an algebra. We expect that the rewriting rules are compatible with the linear structure in the following way. For a rewriting rule

f g relating two elements of an algebra on a ground ﬁeld→ K, then for any scalar λ in K we would like the reduction: λf λg, and for any other element h of the algebra, we would like the following reduction: → f + h g + h.

→

69 5.1. Linear 2-polygraphs

Taken together, these two reductions lead to losing termination of rewriting. Indeed, in that case from the rule f g, we deduce the reductions −f −g and −f + (f + g) −g + (f + g). Finally, we deduce the following reduction → →g f. → As a consequence, the system will never terminate. Further to this remark, it is necessary to adapt the notion of rewriting system to linear situations.→ In the example presented above the reduction −f + (f + g) −g + (f + g) appears as the source of the nontermination problem. There are two ways to solve this problem. The most well-known method is to choose an orientation of the rules induced→ by a monomial order, which is well-founded by deﬁnition, see 5.4.1. This approach is used in various paradigms of linear rewriting as recalled in Chapter 6. In this chapter, we present the categorical description of linear rewriting that extends to associative algebras the notion of 2-polygraph, with an appropriated notion of reduction. The constructions given in this chapter come from [GHM19]. The ground ﬁeld will be denoted by K. We denote by Vect the category of vector spaces over K and linear maps. This category is a monoidal category with the tensor product over K of vector spaces, denoted by ⊗. We will denote by Alg the category of (unital associative) algebras over K.

5.1. LINEAR 2-POLYGRAPHS

We have seen in (2.1.2) that a category can be thought of as a "monoid with several 0-cells". Similarly, the notion of 1-algebroid describes the concept of associative algebra with several 0-cells.

5.1.1. Algebroids. A 1-algebroid over a ground ﬁeld K is a category enriched over the monoidal category Vect. Explicitly, a 1-algebroid A is speciﬁed by the following data:

i) a set A0 of 0-cells, that we will denote by p, q... ii) for every 0-cells p and q, a vector space A(p, q), whose elements are the 1-cells of A, with source p and target q, that we will denote by f, g... iii) for every 0-cells p, q and r, a linear map

?0 : A(p, q) ⊗ A(q, r) − A(p, r)

called the 0-composition of A and whose image on f⊗g is denoted by f?0 g or fg. This composition → is associative, that is the relation:

(f ?0 g) ?0 h = f ?0 (g ?0 h), holds for any 0-composable 1-cells f, g and h, and unitary, that is, for any 0-cell p, there is a 1-cell 1p such that for any 1-cell f in A(p, q), the following relation holds

1p ?0 f = f ?0 1q = f.

A 1-cell f with source p and target q will be graphically represented by

f p / q

70 5.1.2. Remarks

5.1.2. Remarks. A 1-algebra is a 1-algebroid with a single one 0-cell, that can be identiﬁed to an algebras over K. The notion of 1-algebroid was ﬁrst introduced by Mitchell as ring with several objects called K-category in [Mit72], terminology linear category appear also in the literature. A small Z- category is called a ringoid and a one-0-cell ringoid is a ring.

5.1.3. Free 1-algebroid. The free 1-algebroid on a 1-polygraph Λ = (Λ0,Λ1) is the 1-algebroid, de- ` ` noted by Λ1, whose set of 0-cells is Λ0, and for any 0-cells p and q, Λ1(p, q) is the free vector space ∗ ` on Λ1(p, q). In other words, any 1-cell in the space Λ1(p, q) is a linear combination of paths from p to ` q generated by the 1-polygraph Λ. If Λ0 has only one 0-cell, Λ1 is the free algebra with basis Λ1. The ` source and target maps s0 and t0 of the 1-polygraph Λ are extended into maps on Λ1, denoted by s0 and t0, in a natural way making the following two diagrams commutative:

s0 t Λ o Λ` Λ o 0 Λ` 0 c O 1 0 c O 1

ι1 ι1 s0 O t0 O Λ1 Λ1

` where ι1 denotes the inclusion of 1-cells of Λ1 in the free algebroid Λ1.

` 5.1.4. Two-dimensional linear polygraphs. A cellular extension of the 1-algebroid Λ1 is a set Λ2 equipped with two maps s ` o 1 Λ1 o Λ2 t1 ` such that, for every α in Λ2, the pair (s1(α), t1(α)) is a 1-sphere in Λ1, that is, the following globular relations hold s0s1(α) = s0t1(α) and t0s1(α) = t0t1(α). As in the non linear situation of (2.1.10), an element of the cellular extension Λ2 will be graphically represented by a 2-cell with the following globular shape f p α q Õ @ g

` α that relates parallel 1-cells f and g in Λ1, also denoted by f g or by α : f g. We deﬁne a linear 2-polygraph as a triple (Λ0,Λ1,Λ2), where (Λ0,Λ1) is a 1-polygraph and Λ2 is ` a cellular extension of the free 1-algebroid Λ1: ⇒ ⇒

s0 Λ o Λ` 0 c c O 1 c c t0 s0 ι1 s1 O t0 t1 Λ1 Λ2

The elements of Λ2 are called the 2-cells of Λ, or the rewriting rules of Λ.

71 5.1. Linear 2-polygraphs

In the sequel, we will consider polygraphs with one 0-cell denoted •.

5.1.5. The ideal of a linear 2-polygraph. Given a linear 2-polygraph Λ. We denote by I(Λ) the two- ` sided ideal of the free algebra Λ1 generated by the following set of 1-cells

{s1(α) − t1(α) | α ∈ Λ2}. The ideal I(Λ) is made of the linear combinations p λiui(s1(αi) − t1(αi))vi, i=1 X ` for pairwise distinct 2-monomials u1α1v1, . . . , upαpvp of Λ1, and nonzero scalars λ1, . . . , λp.

5.1.6. Presentations of algebras. The algebra presented by a linear 2-polygraph Λ, and denoted by Λ, ` is the quotient of the free algebra Λ1 by the two-sided ideal I(Λ). We denote by f the image of a 1-cell f ` of Λ1 through the canonical projection ` / / π : Λ1 A We say that a linear 2-polygraph Λ is a presentation of an algebra A if the algebra presented by Λ is isomorphic to A. Two linear 2-polygraphs are said to be Tietze equivalent if they present isomorphic algebras.

5.1.7. First toy example. Here our ﬁrst toy example that we will use through this chapter: γ Λ = h x, y, z | xyz x3 + y3 + z3 i.

The free 1-algebroid generated by Λ1 = {x, y, z} is the free algebra Khx, y, zi. The algebra presented by ⇒ the linear 2-polygraph Λ is the quotient of the algebra Khx, y, zi by the two-sided ideal generated by the 1-cell xyz − x3 − y3 − z3.

5.1.8. Other toy examples. We will consider the two following Tietze equivalent linear 2-polygraphs: h x, y | x2 yx i, h x, y | yx x2 i.

5.1.9. 2-algebras. We define a 2-algebra⇒A as an internal 1-category⇒ in the category Alg. Explicitly, it is defined by a diagram t1 oo ? s o 1 × A1 1 / A2 A2 A1 A2 (5.1) i2 where A2 ×A1 A2 is the algebra defined by the following pullback diagram in the category Alg: / A2 ×A1 A2 A2

s1 A2 / A1 . t1

72 5.1.10. Notations

0 0 Elements of the algebra A2 ×A1 A2 are pairs (a, a ) of 1-composable 2-cells a and a , that is satisfying 0 t1(a) = s1(a ). We denote by ab the product of two 2-cells a and b in the algebra A2. The linear structure and the product in the algebra A2 ×A1 A2 are given by setting

(a, a0) + (b, b0) = (a + b, a0 + b0), λ(a, a0) = (λa, λa0), (a, a0)(b, b0) = (ab, a0b0),

0 0 for all pair of 1-composable 2-cells (a, a ) and (b, b ) and scalar λ in K. The morphisms of algebras s1, t1 and ?1 satisfy the axioms in such a way that Diagram (5.1) deﬁnes a 1-category. Explicitly, the following diagrams commute in the category Alg:

i i ? ? 2 / 2 / 1 / 1 / A1 A2 A1 A2 A2 ×A1 A2 A2 A2 ×A1 A2 A2

s1 t π s1 π t id id 1 1 2 1 A1 A1 A2 / A1 A2 / A1 s1 t1

? × id i × id id × i 1 A1 / 2 A1 / o A1 2 A2 ×A1 A2 ×A1 A2 A2 ×A1 A2 A1 ×A1 A2 A2 ×A1 A2 A2 ×A1 A1 ? id ×A1 ?1 ?1 1 π2 π1 ' w A2 ×A A2 / A2 A2 1 ?1 where π1 and π2 denote respectively ﬁrst and second projection.

5.1.10. Notations. For a 1-cell f, the identity 2-cell i2(f) is denoted by 1f, or f if there is no possible 0 0 0 confusion. The 1-composite ?1(a, a ) of 1-composable 2-cells a and a , will be denoted by a ?1 a . Elements of the algebra A1, called 1-cells of A, are graphically pictured as follows

g f f | h # . • or • g /; • h

The elements of A2, called 2-cells of A are graphically represented by

s1(a) $ • a : • . Õ t1(a)

73 5.1. Linear 2-polygraphs

Given 2-cells f g # $ • a ; • and • b : • , Õ Õ f0 g0 the source and target maps s1 and t1 being morphisms of algebras, we have

s1(ab) = s1(a)s1(b), and t1(ab) = t1(a)t1(b), and for any scalars λ and µ in K, we have

s1(λa + µb) = λs1(a) + µs1(b), and t1(λa + µb) = λt1(a) + µt1(b). Hence fg λf + µg # & • ab ; •• λa + µb 8 • Õ Õ f0g0 λf0 + µg0 0 Given 1-cells h, f, f and k in A1 and a 2-cell a in A2 such that f h / k / • • a @ • • Õ f0

0 we will denote by hak : hfk hf k the 0-composite 1h ?0 a ?0 1k.

5.1.11. Properties of 1-composition.⇒ Given 1-composable 2-cells:

f g a b 0 Õ / 0 Õ / • f A • and • g A • 0 0 Õ a Õ b f00 g00

0 0 0 0 in A2 ?A1 A2, the 1-composition ?1 being linear, a ?1 a + b ?1 b is a 2-cell from f + g to f + g and we have

0 0 0 0 (a + b) ?1 (a + b ) = ?1(a + b, a + b ), 0 0 = ?1(a, a ) + ?1(b, b ), 0 0 = a ?1 a + b ?1 b .

0 00 Furthermore, for any scalar λ in K, λ(a ?1 a ) is a 2-cell from λf to λf and we have 0 0 (λa) ?1 (λa ) = λ(a ?1 a ).

74 5.1.12. Remarkable identities in a 2-algebra

0 0 0 0 Finally, the compatibility with the product induces that ?1((a, a )(b, b )) = ?1(ab, a b ). Hence, we have 0 0 0 0 (a ?1 a )(b ?1 b ) = ab ?1 a b . (5.2) Relation (5.2) corresponds to the exchange law in the 2-algebra A between the 1-composition and the product.

5.1.12. Remarkable identities in a 2-algebra. The following properties hold in a 2-algebra A i) for any 1-composable 2-cells a and a0 in A, we have

0 0 a ?1 a = a + a − t1(a), (5.3)

ii) any 2-cell a in A is invertible for the ?1-composition, and its inverse is given by

− a = −a + s1(a) + t1(a). (5.4) iii) for any 2-cells a and b in A, we have

ab = as1(b) + t1(a)b − t1(a)s1(b) = s1(a)b + at1(b) − s1(a)t1(b). (5.5)

0 Relation (5.3) is a consequence of the linearity of the 1-composition ?1. Indeed, for any (a, a ) in A2 ×A1 A2, we have 0 0 0 0 a ?1 a = (a − s1(a ) + s1(a )) ?1 (t1(a) − t1(a) + a ), 0 0 0 = a ?1 t1(a) − s1(a ) ?1 t1(a) + s1(a ) ?1 a , 0 = a − t1(a) + a .

5.1.13. Exercice. Show identities (5.4) and (5.5).

5.1.14. The free 2-algebra on a linear 2-polygraph. The free 2-algebra over a linear 2-polygraph Λ ` ` is the 2-algebra, denoted by Λ2, deﬁned as follows. In dimension 1, it is the free 1-algebra Λ1 over Λ1. ` For dimension 2, we consider the following diagram in the category of Λ1-bimodule t o 1 ` o M Λ1 s1 / Λ2 , i2

` ` M ` ` ` ` where Λ1 is seen as Λ1-bimodule, Λ2 is the Λ1-bimodule Λ1 ⊗ KΛ2 ⊗ Λ1 ⊕ Λ1 and where the linear maps s1, t1 and i2 are deﬁned by:

s1(fαg) = fs1(α)g, t1(fαg) = ft1(α)g and s1(h) = t1(h) = i2(h) = h,

` ` M for all 2-cell α in Λ2, and 1-cells f, g, h in Λ1. The quotient of the Λ1-bimodule Λ2 by the equivalence relation generated by

as1(b) + t1(a)b − t1(a)s1(b) ∼ s1(a)b + at1(b) − s1(a)t1(b),

75 5.1. Linear 2-polygraphs

` ` ` for all a and b in Λ1 ⊗ KΛ2 ⊗ Λ1, has a structure of algebra, denoted by Λ2, and whose product is given by

ab = as1(b) + t1(a)b − t1(a)s1(b).

One proves that the source and target maps are compatible with this quotient, so giving a structure of 2-algebra: t o 1 ` o ` Λ1 s1 / Λ2 . i2

` 5.1.15. Exercise. Let Λ be a linear 2-polygraph. Given 1-cells f and g in Λ1, show that the 1-cell f − g ` belongs to I(Λ) if and only if there exists a 2-cell a : f g in Λ2. As a consequence, the algebra ` ` presented by Λ is obtained by identifying in Λ1 all the 1-cells s1(a) and t1(a), for every 2-cell a in Λ2. ⇒ ` ∗ 5.1.16. Monomials. A monomial in the free 2-algebra Λ2 is a 1-cell of the free monoid Λ1 over Λ1. ` ∗ ` The set monomials of Λ2, also denoted by Λ1, forms a linear basis of the free algebra Λ1. As a conse- ` quence, every nonzero 1-cell f of Λ1 can be uniquely written as a linear combination of pairwise distinct monomials u1, . . . , up:

f = λ1u1 + ... + λpup with λi ∈ K \{0}, for all i = 1, . . . , p. The set of monomials {u1, . . . , up} will be called the support of f and denoted by Supp(f).

` ` 5.1.17. 2-monomials. A 2-monomial of a free 2-algebra Λ2 is a 2-cell of Λ2 with shape uαv, where α ∗ is a 2-cell in Λ2, and u and v are monomials in Λ1:

s1(α) u # v • / • α ; • / • . Õ t1(α)

` ` ` By construction of the free 2-algebra Λ2, and by freeness of Λ1, every non-identity 2-cell a of Λ2 can ` be written as a linear combination of pairwise distinct 2-monomials a1,..., ap and of an 1-cell h of Λ1:

a = λ1a1 + ... + λpap + h. (5.6)

5.1.18. Exercise. Prove that the decomposition in (5.6) is unique up to the following relations

as1(b) + t1(a)b − t1(a)s1(b) = s1(a)b + at1(b) − s1(a)t1(b), (5.7)

` for all 2-monomials a and b in Λ2.

76 5.1.19. Monomial linear 2-polygraphs

5.1.19. Monomial linear 2-polygraphs. A linear 2-polygraph Λ is left-monomial if, for every 2-cell α ∗ of Λ2, the source s1(α) is a monomial in Λ1 \ Supp(t1(α)). Note that a non-left monomial linear 2-polygraph would produce useless ambiguity only due to the linear structure. A linear 2-polygraph Λ is monomial if it is left-monomial and for every 2-cell α of Λ2, t1(α) = 0 holds. A monomial algebra is an algebra admitting a presentation by a monomial linear 2-polygraph.

5.1.20. Exercise. Show that any linear 2-polygraph is Tietze equivalent to a left-monomial linear 2- polygraph.

5.1.21. Examples. The linear 2-polygraph Λ given in Example 5.1.7 is left-monomial. The linear 2- polygraph h x, y | x2 + y2 2xy i is not left-monomial, but it is Tietze equivalent to the following left-monomial 2-polygraph: 0 1 ⇒ Λ0 = h x, y | xy α (x2 + y2) i. 2 The linear 2-polygraphs h x | x2 0 i and h x, y | xy 0 i are monomials. ⇒ ∗ 5.1.22. Degrees and length. For⇒ monomials u and v⇒in Λ1, we denote by degv u the number of different 4 4 occurrences of the monomial v in the monomial u. For instance degx2 x = 3 and degy x = 0. For a ∗ subset M of monomials in Λ1, we denote

degM u = degv u. v∈M X The length of a monomial u in Λ∗, denoted by `(u), is equal to deg u. 1 Λ1

5.2. LINEARREWRITINGSTEPS

5.2.1. Elementary 2-cells. Let Λ be a linear 2-polygraph. An elementary 2-cell of the free 2-algebra ` ` Λ2 is a 2-cell of Λ2 with shape

s1(a) g λ • a • + • /• Õ B

t1(a)

` where a is a 2-monomial, g is a 1-cell of Λ1 and λ is a nonzero scalar in K.

5.2.2. Example. With the polygraph Λ0 of Example 5.1.21, the 2-cell 2xα0y + y3 : 2x2y2 x3y + xy3 − y3 is elementary and the 2-cell ⇒1 xα0 + α0y : x2y + xy2 (x3 + xy2 + x2y + y3) 2 is not elementary. ⇒

77 5.2. Linear rewriting steps

` 5.2.3. Exercise. Show that any 2-cell in a free 2-algebra Λ2 can be decomposed into a 1-composition ` of elementary 2-cells of Λ2

5.2.4. Rewriting steps. Let Λ be a left-monomial linear 2-polygraph. A rewriting step of Λ is an elementary 2-cell u g λ • a • + • /• Õ B

` of Λ2 such that λ is a nonzero scalar and u is not in the support of g.

5.2.5. Examples. For the linear 2-polygraph given in Example 5.1.7, the 2-cell

3xγ − 3xz3 : 3x2yz − 3xz3 %9 3x4 + 3xy3 is a rewriting step. For a linear 2-polygraph having a rule α : u f, the 2-cell

−α + (u + f) : −u + (u + f) %9 −⇒f + (u + f) is not a rewriting step because the monomial u appears in the context u + f.

5.2.6. Exercise. Let Λ be a left-monomial linear 2-polygraph and let a be an elementary 2-cell of the ` ` 2-algebra Λ2. Show that a can be factorised in the 2-algebra Λ2 into

a 3 = b c 4 j~ where b and c are either identities of rewriting steps.

5.2.7. Example. Let Λ be a linear 2-polygraph and let α : u v be a 2-cell of Λ2. The 2-cell −α + (u + v) and α + (5u + 4v) are not rewriting steps of Λ. They can be decomposed respectively as follows: ⇒

−α + (u + v) α + (5u + 4v) −u + (u + v) $8 −v + (u + v) u + (5u + 4v) $8 v + (5u + 4v) = = v α 3 k 6α + 4v 5α + 5v (1 − 1)u + v !5 10v j~

78 5.2.8. Rewriting sequences

` 5.2.8. Rewriting sequences. A 2-cell a of Λ2 is positive, or a rewriting sequence, if it is an identity or a 1-composite a1 ak f0 f1 ··· fk−1 fk of rewriting steps of Λ. ⇒ ⇒ ⇒ ⇒ ` 5.2.9. Reduced cells. A 1-cell f of Λ1 is called reduced, or irreducible, with respect to Λ2, if there is no rewriting step of Λ with source f. As a consequence, a 1-cell is reduced if and only if it is the zero ` ∗ ` 1-cell of Λ1, or a linear combination of reduced monomials in Λ1. The reduced 1-cells of Λ1 form a ` nf ∗ vector subspace of Λ1, denoted by Λ1 . Since Λ is left-monomial, the set of reduced monomials of Λ1, irm nf denoted by Λ1 , forms a basis of the vector space Λ1 . We denote by s1(Λ) the set of redex of a reduced left-monomial linear 2-polygraph Λ deﬁned by

s1(Λ) = {s1(α) | α in Λ2}.

In [Ani86], a redex is called an obstruction. The number of possible application of rules of Λ2 to a monomial u is deg u. s1(Λ)

5.2.10. Reduced linear 2-polygraphs. We say that a linear 2-polygraph Λ is left-reduced if, for every 2-cell α in Λ2, the 1-cell s1(α) is reduced with respect to Λ2 \{α}. We say that Λ is right-reduced if, for every 2-cell α of Λ, the 1-cell t1(α) is reduced. The linear polygraph Λ is reduced if it is both left-reduced and right-reduced.

5.2.11. Exercise. Show that any left-monomial linear 2-polygraph is Tietze equivalent to a reduced left-monomial linear 2-polygraph.

` 5.2.12. Normal forms. If f is a 1-cell of Λ1, a normal form for f with respect to Λ2 is a reduced 1-cell g ` ` of Λ1 such that there exists a positive 2-cell a : f g in Λ2.

5.3. TERMINATION FOR⇒ LINEAR 2-POLYGRAPHS

We recall the notion of rewrite relation for linear 2-polygraphs from [GHM19]. Let us ﬁx a left-monomial linear 2-polygraph Λ.

5.3.1. Termination. The rewrite relation of Λ is the binary relation, denoted by ≺Λ on the set of ∗ monomial Λ1 deﬁned by

i) w ≺Λ u for every 2-cell α : u f of Λ2 and every monomial w in Supp(f), ii) u0 ≺ u implies vu0w ≺ vuw for all monomials u, u0, v and w of Λ∗. Λ Λ ⇒ 1 We say that Λ terminates if its rewrite relation ≺Λ is wellfounded, that is, there is no inﬁnite de- ∗ scending chains in Λ1:

u1 Λ u2 Λ u3 Λ ... Λ un Λ un+1 Λ ...

79 5.4. Monomial orders

α 2 2 2 5.3.2. Example. Consider the linear 2-polygraph Λ = h x, y | xy x + y i. We have x ≺Λ xy and 2 y ≺Λ xy. It follows that 2 2 2 x y Λ xy Λ x y. ⇒

Hence the relation ≺Λ is not a wellfounded and the polygraph is not terminating. Note that, we have an inﬁnite sequence of rewriting steps: x3 + αy xα x2y x3 + xy2 %9 x3 + y3 + x2y ...

⇒ ⇒ ` 5.3.3. The rewrite relation on 1-cells. The rewrite relation ≺Λ is extended to the 1-cells of Λ1 by setting, for any 1-cells f and g, g ≺Λ f if the following two conditions hold i) there exists a monomial w in Supp(f) which is not in Supp(g), ii) for any monomial v in Supp(g)\ Supp(f), there exists a monomial u in Supp(f)\ Supp(g), such that v ≺Λ u

5.3.4. Proposition. The rewrite relation ≺Λ is wellfounded on 1-cells if and only if it is wellfounded on monomials.

If Λ terminates, then for every rewriting step a of Λ, we have t1(a) ≺Λ s1(a). This implies that the ` 2-algebra Λ2 contains no inﬁnite sequence of pairwise 1-composable rewriting steps

a1 ak f0 f1 ··· fk−1 fk ··· ` so that every 1-cell of Λ admits at least one normal form with respect to Λ2. 1 ⇒ ⇒ ⇒ ⇒ ⇒

5.4. MONOMIALORDERS

∗ 5.4.1. Monomial orders. A total order ≺ on the set of monomials Λ1 is a monomial order if the following conditions are satisﬁed ∗ i) ≺ is a well-order, that is, there is no inﬁnite descending chains in Λ1.

u1 u2 u3 ... un un+1 ...

ii) ≺ is compatible with the multiplicative structure on monomials, that is u ≺ u0 implies vuw ≺ vu0w,

0 ∗ for all monomials u, u , v and w in Λ1.

5.4.2. Example. Given a total order relation ≺ on Λ1, we define the left degree-wise lexicographic ∗ order generated by ≺, or deglex order generated by ≺, as the order ≺deglex on Λ1 that compare two monomials first by degree and then lexicographically. It is defined by

i) y1 . . . yp ≺deglex x1 . . . xq, if p < q,

ii) y1 . . . yj−1yj . . . yp ≺deglex y1 . . . yj−1xj . . . xp, if yj ≺ xj.

80 5.4.3. Exercise

5.4.3. Exercise. Show that the order ≺deglex is a monomial order.

5.4.4. Exercise. Explain why the pure lexicographic order is not a monomial order. Show that it is neither a well-order nor compatible with the product of monomials.

5.4.5. Polygraph compatible with a monomial order. A linear 2-polygraph Λ is say to be compatible with a monomial order ≺ if for every 2-cell α : u f of Λ2, then w ≺ u for any monomial w in the support of f. The monomial order ≺ is thus a well-founded rewrite relation for Λ. It follows that any linear 2-polygraph compatible with a monomial order⇒ is terminating. The converse is false in general as we will see in Exercise 5.4.7.

α 5.4.6. Example. Consider the linear 2-polygraph Λ = h x, y | x2 xy − y2 i. It is Tietze equivalent to the linear 2-polygraph of Example 5.3.2, but it is terminating. Indeed, having xy ≺ x2 and y2 ≺ x2, the linear 2-polygraph Λ is compatible with the deglex order ≺deglex⇒ induced by y ≺ x, hence it is terminating. An other way to prove that Λ is terminating, is to count the number of occurrence of x in ∗ monomials. For any u in Λ1, let denote by A(u) the number of occurrence of x in u. To prove that the linear 2-polygraph Λ terminates, it is sufﬁcient to check that, for every rewriting step a : s1(a) f, we have A(s1(a)) > A(v), for any monomial v in Supp(f). ⇒ 5.4.7. Exercise. Show that the linear 2-polygraph Λ given in Example 5.1.7 is terminating. Show that Λ is not compatible with a monomial order.

5.4.8. Exercise, [Ber78, Exercice 5.2.1.]. Examine termination of the linear 2-polygraph h x, y | α i in each of the following situations x2y α yx, yx α x2y, x2y2 α yx, yx α x2y2.

5.4.9. Noetherian induction.⇒ Let us recall⇒ the principle of⇒ noetherian induction⇒ for terminating rewriting systems, see [Hue80] for more details. Let Λ be a left-monomial terminating linear 2-polygraph. ` ` Given a property P(f) of the 1-cells f of Λ1. In order to show that P(f) holds for any 1-cell f of Λ1, it sufﬁces to show that

i) P(f) holds for f reduced with respect to Λ2,

ii) P(f) holds under the assumption that P(g) is hold for every g ≺Λ f.

` ` 5.4.10. Leading terms. Let Λ1 be a free algebra over a set Λ1 and let ≺ be a monomial order on Λ1. ` For a nonzero 1-cell f of Λ1, the leading monomial of f with respect to ≺ is the monomial of f, denoted by lm(f), such that w ≺ lm(f), for any monomial w in the support of f. The leading coefficient of f is ` the coefficient lc(f) of lm(f) in f, and the leading term of f is the 1-cell lt(f) = lc(f) lm(f) of Λ1. We also define lt(0) = lc(0) = lm(0) = 0. ` Note that for any 1-cells f and g in Λ1, we have f ≺ g if and only if either lm(f) ≺ lm(g) or (lm(f) = lm(g) and f − lt(f) ≺ g − lt(g)). The following property

lt(fg) = lt(f) lt(g),

81 5.5. Conﬂuence and convergence for any 1-cells f and g is also useful.

` ` 5.4.11. Leading polygraph. Given a monomial order ≺ on Λ1 and a nonzero 1-cell g in Λ1, we deﬁne the 2-cell: 1 α : lm(g) %9 lm(g) − g. g,≺ lc(g) ` For any set G of nonzero 1-cells in Λ1, the leading 2-polygraph associated to G with respect to ≺ is the linear 2-polygraph Λ(G, ≺) whose set of 1-cells is Λ1 and

Λ(G, ≺)2 = {αg,≺ | g ∈ G}. By deﬁnition, the leading polygraph Λ(G ≺) is compatible with the monomial order ≺. ∗ A monomial w in Λ1 is G-reduced with respect to the monomial order ≺ if it reduced with respect ∗ to Λ(G, ≺)2, that is, there is no factorisation w = u lm(g)v, with u and v monomials in Λ1 and g in G. A set G of 1-cells is reduced with respect to the monomial order ≺ if for any 1-cell g in G, any monomial in the support of g is (G \{g})-reduced.

5.5. CONFLUENCEANDCONVERGENCE

` 5.5.1. Suppose that Λ is a terminating left-monomial linear 2-polygraph. Every 1-cell f of Λ1 admits ` at least a normal form fe. That is, feis reduced and there exists a positive 2-cell a : f fein Λ2. As a consequence, we have a decomposition ⇒ f = fe+ (f − fe),

nf ` with fein Λ1 and f−fein I(Λ) by Exercice 5.1.15. It follows that the vector space Λ1 admits the following decomposition ` nf Λ1 = Λ1 + I(Λ). (5.8) In this section we show that the decomposition (5.8) is direct if and only if the polygraph Λ is conﬂuent.

5.5.2. Example. Note that the decomposition (5.8) is not direct in general. Indeed, consider the linear β 2-polygraph Λ = h x, y | x2 xy i. It is terminating thanks to the deglex order generated by x > y. Consider the two following reduction sequences reducing the 1-cell x3: ⇒ βx (< xyx x3

xβ "6 x2y %9 xy2 βy

Thus the 1-cell xyx − xy2 = −(x2 − xy)x + x(x2 − xy) + (x2 − xy)y nf nf is both in Λ1 and I(Λ). It follows that the sum Λ1 + I(Λ) is not direct.

82 5.5.3. Branchings and conﬂuence

5.5.3. Branchings and conﬂuence. Let Λ be a left-monomial linear 2-polygraph. A branching of Λ ` is a non-ordered pair (a, b) of positive 2-cells of Λ2 with a common source s1(a) = s1(b). A branching (a, b) is local if both a and b are rewriting steps of Λ. A branching (a, b) of Λ is conﬂuent if there exist positive 2-cells a0 and b0 of Λ as in the following diagram

0 a '; g a * f f0 6J b #7 h b0 We say that Λ is confluent (resp. locally confluent) if every branching (resp. local branching) of Λ is ` confluent. An immediate consequence of the confluence property is that every 1-cell of Λ1 admits at most one normal form. 5.5.4. Proposition. Let Λ be a terminating left-monomial linear 2-polygraph. The following conditions are equivalent. i) Λ is confluent.

ii) Every 1-cell of I(Λ) admits 0 as a normal form with respect to Λ2. ` ` nf iii) The vector space Λ1 admits the direct decomposition Λ1 = Λ1 ⊕ I(Λ). ` Proof. i) ii). Let f be a 1-cell in the ideal I(Λ), then there exists a 2-cell a : f 0 in Λ2. The polygraph Λ being conﬂuent, the 1-cells f and 0 have the same normal form. Finally, 0 beaing reduced, this implies⇒ that 0 is a normal form for f. ⇒ nf nf ii) iii). Prove that Λ1 ∩ I(Λ) = 0. If f is in Λ1 , then f is reduced and, thus, admits itself as nf normal form. If f is in I(Λ), then f admits 0 as a normal form by ii). Hence Λ1 ∩ I(Λ) = 0. iii)⇒ i). Consider a branching (a, b) of Λ with a : f g and b : f h. Since Λ terminates, each ` of g and h admits at least one normal form. Hence, there exist positive 2-cells a1 and b1 in Λ2:

⇒ a1 ⇒ ⇒ a &: g %9 g1 f

b $8 h %9 h1 b1 − with g1 and h1 reduced. It follows that g1 −h1 is also reduced. Moreover, the 2-cell (a?1 a1) ?1 (b?b1) nf has g1 as source and h1 as target. This implies that g1 − h1 is also in I(Λ). As Λ1 ∩ I(Λ) = 0, we have g1 − h1 = 0, hence the branching (a, b) is conﬂuent.

5.5.5. Convergence. We say that a left-monomial linear 2-polygraph Λ is convergent if it terminates ` and it is confluent. In that case, every 1-cell f of Λ1 has a unique normal form, denoted by fb, such ` that f = g holds in Λ if and only if fb= gb holds in Λ1. As a consequence, if Λ is a convergent presentation of an algebra A, the assignment of every 1-cell f ` ` of A to the normal form fb, defines a section ι : A − Λ2 of the canonical projection π : Λ − A. The section ι is a linear map, i.e., it satisfies λf\+ µg = λfb+ µgb, and it preserves the identities because Λ terminates. → →

83 5.5. Conﬂuence and convergence

5.5.6. Exercise. Show that the section ι is not a morphism of algebras in general.

irm 5.5.7. Theorem. Let A be an algebra and Λ be a convergent presentation of A. The set Λ1 of reduced nf monomials is a linear basis of A. Moreover, the vector space Λ1 equipped with the product deﬁned by nf f · g = fgb , for any 1-cells f and g in Λ1 , is an algebra isomorphic to A. Proof. Suppose that Λ is a convergent linear 2-polygraph. By Proposition 5.5.4 the following sequence of vector spaces is exact:

/ / / ` / / nf / 0 I(Λ) Λ1 Λ1 0

nf irm irm The vector space Λ1 admits Λ1 as a basis, hence Λ1 forms a basis of the vector space underlying ` ` the quotient algebra Λ1/I(Λ), that is the algebra A. The polygraph Λ being convergent, any 1-cell of Λ1 has a unique normal form, hence the product deﬁned by f · g = fgb is associative. Indeed, for any 1-cells f, g and h, we have

(f · g) · h = fgb · h = fghdb = fdghc = f · ghc = f · (g · h).

nf nf It follows that this product equips Λ1 with a structure of algebra in such a way that Λ1 is isomorphic to the algebra A.

5.5.8. Exercise. Compute a linear basis of the algebra presented by hx, y | xy = x2i.

5.5.9. Exercise. Compute a linear basis for the symmetric algebra on k variables presented by

τij h x1, . . . , xk | xixj xjxi | 1 6 i < j 6 k i and for the skew-polynomial algebra on k variables⇒ presented by

τij j h x1, . . . , xk | xixj qixjxi | 1 6 i < j 6 k i,

j where the qi are scalars in K. ⇒

5.5.10. Exercise: Poincaré-Birkhoff-Witt theorem, [Bok76, §1.], [Ber78, Theorem 3.1]. Consider an ordered bases x1 ≺ x2 ≺ ... ≺ xk of a Lie algebra g. Consider the following ideals of the free tensor algebra T(g) over g:

I = h xjxi − xixj | 1 6 i < j 6 k i, J = h xjxi − xixj + [xi, xj] | 1 6 i < j 6 k i.

Show that the symmetric algebra Sg = T(g)/I and the enveloping algebra Ug = T(g)/J are isomorphic as vector spaces.

84 5.5.11. From local to global conﬂuence

5.5.11. From local to global confluence. The Newman lemma, also called the diamond lemma, states that for terminating rewriting systems local confluence and confluence are equivalent properties. This result was proved by Newman in [New42] for abstract rewriting systems. A short and simple proof of this result was given by Huet in [Hue80] using the principle of noetherian induction. Let us recall the arguments of this proof for linear 2-polygraphs.

5.5.12. Theorem (Newman’s Lemma). Let Λ be a terminating left-monomial linear 2-polygraph. Then Λ is conﬂuent if and only if it is locally conﬂuent.

Proof. One implication is trivial. Suppose Λ locally confluent and prove that it is confluent at every 1- ` cell f of Λ1. We proceed by noetherian induction on f. If f is reduced, the only branching with source f is (1f, 1f) which is confluent. ` Suppose that f is a nonreduced 1-cell of Λ1 and such that Λ is confluent at every 1-cell g ≺ f. Con- sider a branching (a, b) of Λ with source f. If a or b is an identity, then (a, b) is confluent. Otherwise, we prove that the branching (a, b) is confluent by induction. Since a and b are not identities, they admit decompositions a = a1 ?1 a2 and b = b1 ?1 b2 where a1 and b1 are rewriting steps, and a2 and b2 are positive 2-cells. By local confluence, the local branching (a1, b1) is confluent. Hence there exist positive 0 0 2-cells a1 and b1 as indicated in the following diagram

f a1 b1

Local ' g s× h 1 conﬂuence 1 a2 b2 0 0 a1 b1 { 0 0 qÕ g Induction f1 h

0 0 c a2 3 g h| Induction 0 b2 d "6 f0 mÑ

0 We have g1 ≺Λ f and h1 ≺Λ f. Then we apply the induction hypothesis on the branching (a2, a1) to get 0 0 0 positive 2-cells a2 and c, and, then, to the branching (b1 ?1 c, b2) to get positive 2-cells d and b2, which complete the proof.

5.6. THE CRITICAL BRANCHINGS THEOREM

5.6.1. Local branchings. A case analysis leads to a partition of the local branchings of a left-monomial linear 2-polygraph Λ into the following four families:

` ` i) Aspherical branchings, for all 2-monomial a : u f of Λ2, nonzero scalar λ, and 1-cell h of Λ1

⇒

85 5.6. The Critical Branchings Theorem

such that the monomial u is not in the support of h:

λa + h

+ λu + h λf + h 4H

λa + h

` ii) Additive branchings, for all 2-monomials a : u f and b : v g of Λ2, nonzero scalars λ and µ, ` and 1-cell h of Λ1 such that the monomials u and v are not in the support of h: ⇒ ⇒ λa + µv + h '; λf + µv + h

λu + µv + h

λu + µg + h λu + µb + h #7

` iii) Peiffer branchings, for all 2-monomials a : u f and b : v g of Λ2, nonzero scalar λ, and ` 1-cell h of Λ1 such that the monomial uv is not in the support of h: ⇒ ⇒ λav + h %9 λfv + h λuv + h λug + h λub + h $8

` iv) Overlapping branchings, for all 2-monomials a : u f and b : u g of Λ2 such that the ` branching (a, b) is neither aspherical nor Peiffer, and all nonzero scalar λ and 1-cell h of Λ1 such that the monomial u is not in the support of h: ⇒ ⇒

λa + h %9 λf + h λu + h λg + h λb + h %9

5.6.2. Critical branchings. A critical branching of a left-monomial linear 2-polygraph Λ is an overlapping branchings, as deﬁned in 5.6.1, with λ = 1 and h = 0, and that is minimal for the relation on branchings deﬁned by

0 0 0 ∗ (a, b) v (waw , wbw ) for any w and w in Λ1.

86 5.6.3. Exercise

By case analysis on the source of critical branchings, they must have one of the following two shapes

EY EY α α / / / / / / F B β β Õ Õ with α, β in Λ2. When the linear 2-polygraph Λ is reduced, the ﬁrst case cannot occur since, otherwise, the monomial s1(α) would be reducible by β.

5.6.3. Exercise. Let Λ be a reduced linear 2-polygraph. Show that for any critical branching

EY α / v / w / u B β Õ the monomial u, v and w are reduced and cannot be identities or null.

5.6.4. Critical branching lemma. By the Newman lemma 5.5.12, for terminating rewriting systems, local confluence and confluence are equivalent properties. It turns out that one can decide whether a rewriting system is convergent by checking local confluence. For string rewriting systems, that is 2- polygraphs, the critical branching lemma states that local confluence is equivalent to the confluence of all critical branching, see [GM18, 3.1.5] for details. For linear 2-polygraphs the critical branching lemma given in [GHM19] differs from the case of 2-polygraphs. Indeed, in the linear setting the termination hypothesis is required. Moreover, nonoverlapping branchings may be non confluent as illustrated by the following example in which an additive branching is nonconfluent.

5.6.5. Example. Some local branchings can be nonconfluent without termination, even if critical confluence holds. Indeed, consider for instance the following linear 2-polygraph β h x, y, z, t | xy α xz, zt 2yt i has no critical branching, but it has a nonconfluent additive branching: ⇒ ⇒ 4αt 4xβ 2xβ )= 4xyt %9 4xzt %9 ···

αt + xzt )= 2xzt xzt + xβ , xyt + xzt = xzt + 2xyt 2F αt + 2xyt xyt + xβ !5 3xyt

3αt !5 3xzt %9 6xyt %9 ··· 3xβ 6αt

87 5.6. The Critical Branchings Theorem

5.6.6. If a linear 2-polygraph Λ is terminating and with any critical branching confluent, we can show that such an additive branching is confluent by noetherian induction on the sources of the branchings. Let consider an additive branching (λu+µv+h, λu+µg+h) as in (5.6.1) and suppose that Λ is locally confluent at every g ≺Λ λu + µv + h. By linearity of the 1-composition, the following equation

(λa + µv + h) ?1 (λf + µb + h) = (λu + µb + h) ?1 (λa + µg + h)

` holds in the free 2-algebra Λ2:

0 a1 λf + µv + h !5 f0 0 λa + µv + h *> /C a2 = λf + µb + h c 2 Ù λu + µv + h = λf + µg + h Induction k ,@ AU λa + µg + h = d λu + µb + h 4 / 0 b0 λu + µg + h )= g 2 0 b1

Note that the dotted 2-cells λa + µg + h and λf + µb + h may be not positive in general. Indeed, the monomial u can be in the support of g or the monomial v can be in the support of f, as illustrated in Example 5.6.5. However, those 2-cells are elementary, hence there exist, see Exercise 5.2.6, positive 0 0 2-cells a1, b1, c and d that satisfy

0 0 a1 = (λf + µb + h) ?1 c and b1 = (λa + µg + h) ?1 d.

We have f ≺Λ u and g ≺Λ v, hence λf + µg + h ≺Λ λu + µv + h. Thus, the branching (c, d) is 0 0 confluent by induction hypothesis, yielding the positive 2-cells a2 and b2. In this way, one shows that under terminating hypothesis, all local branching given in (5.6.1) are confluent if all critical branching are confluent.

5.6.7. Theorem (Critical branching lemma). A terminating left-monomial linear 2-polygraph is locally conﬂuent if and only if all its critical branchings are conﬂuent.

As consequence of the critical branching lemma and of the Newman lemma 5.5.12, a terminating left-monomial linear 2-polygraph is conﬂuent if all its critical branchings are conﬂuent. In particular a terminating left-monomial 2-polygraph with no critical branching is convergent.

5.6.8. Example. The linear 2-polygraph given in Example 5.1.7 is terminating, see Exercise 5.4.7. Moreover, it does not have critical branching, hence it is convergent.

5.6.9. The Knuth-Bendix completion procedure. The completion procedure for terminating 2-polygraphs given in (2.5.1) can be adapted to linear 2-polygraphs as follows. Let Λ be a left-monomial linear 2- ∗ polygraph compatible with a monomial order ≺ on Λ1.A Knuth-Bendix completion of Λ is a linear

88 5.6.10. Exercice

2-polygraph KB(Λ) obtained by the following procedure that examines the conﬂuence of the set of critical branchings.

∗ Input: Λ be a left-monomial linear 2-polygraph compatible with a monomial order ≺ on Λ1. KB(Λ):=Λ Cb:={ critical branchings with respect to Λ2 } while Cb 6= ∅ do Picks a branching in Cb: f $8 v u g &: w Cb := Cb \{(f, g)} Reduce v to a normal form bv with respect to KB(Λ)2 Reduce w to a normal form wb with respect to KB(Λ)2 f $8 v %9 bv u g %9 w %9 wb

g = bv − wb if g 6= 0 then 1 KB(Λ)2 := KB(Λ)2 ∪ { αg,≺ : lm(g) lm(g) − lc(g) g } Cb := Cb ∪ { critical branching created by αg,≺ } end ⇒ end If the procedure stops, it returns a ﬁnite convergent left-monomial linear 2-polygraph KB(Λ). Other- wise, it builds an increasing sequence of left-monomial linear 2-polygraphs, whose limit is also denoted by KB(Λ). Note that, if the starting linear 2-polygraph Λ is convergent, then the Knuth-Bendix completion of Λ is Λ itself. The linear 2-polygraph KB(Λ) obtained by this procedure depends on the order of examination of the critical branchings. Finally, since all the operations of adding new rules performed by the procedure are Tietze transformations, the linear 2-polygraph KB(Λ) is Tietze-equivalent to Λ.

5.6.10. Exercice. Prove that the following linear 2-polygraph has a nonconﬂuent Peiffer branching

β h x, y, z | xy α 2x, yz z i.

5.6.11. Weyl algebras. Let K be a ﬁeld of characteristic⇒ zero.⇒ The Weyl algebra of dimension n over K is the algebra presented by the linear 2-polygraph whose 1-cells are

x1, . . . , xn, ∂1, . . . , ∂n and with the following 2-cells:

xixj xjxi, ∂i∂j ∂j∂i, ∂ixj xj∂i, for any 1 6 i < j 6 n,

⇒ ⇒ ⇒

89 5.7. Coherent presentations of algebras

∂ixi xi∂i + 1, for any 1 6 i 6 n. This polygraph is convergent with the following six families of conﬂuent critical branchings: ⇒

)= xjxixk %9 xjxkxi *> ∂j∂i∂k %9 ∂j∂k∂i ! " xixjxk xkxjxi ∂i∂j∂k ∂k∂j∂i ;O ;O !5 xixkxj %9 xkxixj !5 ∂i∂k∂j %9 ∂k∂i∂j

)= xj∂ixk %9 xjxk∂i )= ∂j∂ixk %9 ∂jxk∂i " " ∂ixjxk xkxj∂i ∂i∂jxk xk∂j∂i ;O ;O

!5 ∂ixkxj %9 xk∂ixj !5 ∂ixk∂j %9 xk∂i∂j

,@ xi∂ixj + xj %9 xixj∂i + xj (< ∂j∂ixj %9 ∂jxj∂i ( ( ∂ixjxk xjxi∂i + xj ∂i∂jxj xj∂j∂i + ∂i Ui 4H 2 "6 ∂ixjxi %9 xj∂ixi ∂ixj∂j + ∂i %9 xj∂i∂j + ∂i where 1 6 i < j 6 n.

5.6.12. Exercice. In his seminal paper on the diamond lemma, Bergman point out that he was ﬁrst led to the ideas of his paper with the following American Mathematical Monthly Advanced Problem 5082, [Ber78, 2.1.].

Let R be a ring in which, if either x + x = 0 or x + x + x = 0, it follows that x = 0. Suppose that a, b, c and a + b + c are all idempotents in R. Does it follows that ab = 0?

Solve this problem. Hints. Consider the following linear 2-polygraph:

Λ = h a, b, c | a2 a, b2 b, c2 c, ba −ab − bc − cb − ac − ca i.

1/ List all critical branchings of⇒Λ. 2/ Compute⇒ a⇒ convergent⇒ left-monomial linear 2-polygraph KB(Λ) by applying the Knuth-Bendix completion procedure to Λ. 3/ List all irreducible monomials with respect to KB(Λ)2. 4/ Conclude that ab 6= 0.

5.7. COHERENT PRESENTATIONS OF ALGEBRAS

In this last section, we recall from [GHM19] the notion of coherent presentation for an algebra as a presentation of the algebra extended by a family of generating syzygies. We explain how to generate syzygies when the presentation is convergent.

90 5.7.1. Linear 3-polygraph

` 5.7.1. Linear 3-polygraph. Let Λ be a linear 2-polygraph. A cellular extension of the free 2-algebroid Λ2 is a set Λ3 equipped with maps s ` o 2 Λ2 o Λ3 t2 ` such that, for every F in Λ3, the pair (s2(F), t2(F)) is a 2-sphere in Λ2, that is, s1s2(F) = s1t2(F) and ` t1s2(F) = t1t2(F) hold in Λ2. The elements of Λ3 are the 3-cells of the cellular extension and graphically represented by s2(F)

f F &g Õ 6J

t2(F)

A linear 3-polygraph is a data (Λ0,Λ1,Λ2,Λ3), where (Λ0,Λ1,Λ2) is a linear 2-polygraph and Λ3 ` is a cellular extension of the free 2-algebroid Λ2:

s0 s1 Λ oo Λ` o Λ` 0 c c O 1 c c O 2 c c t0 t1 s0 ι1 s1 ι2 s2 O O t0 t1 t2 Λ1 Λ2 Λ3

5.7.2. Three-dimensional algebras. We deﬁne a 3-algebra as an internal 2-category in the category Alg: s s o 1 o 2 A1 o A2 o A3 t1 t2

In particular, the algebras A1 and A2 with composition ?1 : A2 ×A1 A2 A2 form a 2-algebra. The 3-cells can be composed in two different ways: → ?1 : A3 ×A1 A3 A3 ?2 : A3 ×A2 A3 A3 by ?1 along their 1-dimensional boundary, and by ?2 along their 2-dimensional boundary as pictured → → in (3.2.5). The source and target maps s1, s2 and t1, t2 being morphisms of algebras, the product of 3-cells F and G satisﬁes: f g fg

F G FG 0 0 0 0 • a %9 a B • b %9 b B • 7− • ab %9 a b > • Õ Õ Õ Õ Õ Õ → f0 g0 f0g0

These compositions and the product satisfy remarkable properties similar to those given in (5.1.12) for 2-algebras.

91 5.7. Coherent presentations of algebras

5.7.3. Free 3-algebras. The free 3-algebra over a linear 3-polygraph Λ is constructed similarly to the ` free 2-algebra given in (5.1.14). It is the 3-algebra, denoted by Λ3, whose underlying 2-algebra is the ` free 2-algebra Λ2, and its 3-cells are all the formal 1-composition, 2-composition and product of 3-cells of Λ3, of identities of 2-cells, up to associativity, identity, exchange and inverse relations, see [GHM19] for more details.

5.7.4. Coherent presentations of algebras. A coherent presentation of an algebra A is a linear 3- polygraph Λ such that

i) the linear 2-polygraph (Λ0,Λ1,Λ2) is a presentation of A,

` ii) Λ3 is a homotopy basis of the free 2-algebra Λ2, that is, a cellular extension

s ` o 2 Λ2 o Λ3 t2

` such that for every 2-sphere (a, b) of the free 2-algebra Λ2, there exists a 3-cell A in the free ` 3-algebra Λ3 such that s2(A) = a and t2(A) = b.

5.7.5. Squier’s completion. Let Λ be a left-monomial linear 2-polygraph. Suppose that all critical branching of Λ are conﬂuent. For every critical branching (a, b) in Λ, we choose two positive 2-cells a0 and b0 making the branching conﬂuent:

0 a (< g a

( 0 (5.7.6) f F(a,b) f Õ 8L b "6 h b0

0 0 For any such a conﬂuent branching, we consider a 3-cell F(a,b) : a ?1 a V b ?1 b . The set of such 3-cells

Λ3 = { F(a,b) | (a, b) is a critical branching }

` forms a cellular extension of the free 2-algebra Λ2. The linear 3-polygraph (Λ0,Λ1,Λ2,Λ3) is a Squier’s completion of Λ. When the polygraph is conﬂuent, there exists such a Squier’s completion. However, the cellular extension Λ3 is not unique in general. Indeed, the 3-cells can be directed in the reverse way and a branching (a, b) can have several possible positive 2-cells a0 and b0 making the branching conﬂuent. The following result is a formulation of the Squier Lemma, [SOK94], in the setting of linear 2- polygraphs.

5.7.7. Theorem ([GHM19, Thm. 4.3.2]). Let A be an algebra and let Λ be a convergent left-monomial presentation of A. Any Squier’s completion of Λ is a coherent presentation of A.

92 5.7.8. Linear oriented syzygies

5.7.8. Linear oriented syzygies. Let Λ be presentation of an algebra A. Any nontrivial 2-sphere (a, b) ` in the free 2-algebra Λ2 is called a linear oriented 3-syzygy of the presentation Λ. If Λ is extended into ` a coherent presentation (Λ, Λ3) of the algebra A, the quotient 2-algebra Λ2/Λ3 is aspherical, that is, ` for any 2-sphere (a, b) in Λ2/Λ3, we have a = b. In other words, the cellular extension Λ3 forms a generating set of linear 3-syzygies of the presentation Λ. Theorem 5.7.7 say that, when the presentation Λ is convergent the 3-cells deﬁned by conﬂuence diagrams of the critical branchings, as in (5.7.6), form a family of generator for 3-syzygies.

5.7.9. Exercice. Let {F1,...,Fk} be a generating set for linear 3-syzygies of a linear 2-polygraph Λ. − − Prove that {F1 ,...,Fk } is also a generating set for linear 3-syzygies of Λ.

α 5.7.10. Example. The linear 2-polygraph h x | x2 0 i has one critical branching

αx ⇒ ' x3 F 0 Õ 7K xα which is conﬂuent. The polygraph being convergent the 3-cell F : αx V xα generates all linear 3- syzygies of this presentation.

5.7.11. Example. Consider the algebra A presented by the linear 2-polygraph

γ Λ = h x, y, z | xyz x3 + y3 + z3 i given in Example 5.1.7. It does not have critical branching, hence any Squier’s completion of Λ is empty. ⇒ As a consequence, Λ can be extended into a coherent presentation with an empty homotopy basis. That is, there is no 3-syzygy for this presentation. The linear 2-polygraph h x, y, z | αf, β i considered in Example 6.3.7 is Tietze equivalent to Λ, convergent and compatible with a monomial order. It has three critical branchings, as shown in Example 6.3.7. It can be extended into a coherent presentation of A with three generating 3-syzygies.

5.7.12. Exercise. Give an explicit description of the 3-cells of a coherent presentation on the linear 2-polygraph Λ0 of Example 5.7.11.

5.7.13. Exercise. Compute a coherent presentation for the algebras presented by the following linear 2-polygraphs 1) h x, y | xyx y2 i.

α 2 β −1 2 2) h x, y, z | yz⇒ −x , zy −λ x i, where λ ∈ K \{0, 1}, see [PP05, 4.3].

5.7.14. Exercise.⇒ Compute⇒ a minimal coherent presentation for the algebra presented by the linear 2- polygraph h x | x3 = 0 i.

93 5.7. Coherent presentations of algebras

94 CHAPTER 6

Paradigms of linear rewriting

Contents 6.1 Composition Lemma ...... 96 6.2 Reduction operators ...... 98 6.3 Noncommutative Gröbner bases ...... 99

In this chapter, we survey several approaches in linear rewriting. The most well-known is given by the Gröbner basis theory for ideals in commutative polynomial rings introduced by Buchberger in [Buc65]. A subset G of an ideal I in the polynomial ring K[x] of commutative polynomials is a Gröbner basis of I with respect to a given monomial order ≺, if the leading term ideal of I is generated by the set of leading monomials of G, that is h lt≺(I) i = h lt≺(G) i. Buchberger introduced the notion of S-polynomial to describe the obstructions to local confluence and gave an algorithm for computation of Gröbner bases, [Buc65, Buc06], see also [Buc87] for an historical account. Any ideal I of a commutative polynomial ring K[x] has a finite Gröbner basis. Indeed, the Buchberger algorithm on a finite family of generators of an ideal I always terminates and returns a Gröbner basis of the ideal I. Shirshov introduced in [Shi62] an algorithm to compute a linear basis of a Lie algebra defined by generators and relations. He used the notion of composition of elements in a free Lie algebra, that corresponds to the notion of S-polynomial in the work of Buchberger. He gave an algorithm to compute bases in free algebras having the computational properties of the Gröbner bases. He proved that irreducible elements for such a basis forms a linear basis of the Lie algebra. This result is called now the Composition Lemma for Lie algebras.

95 6.1. Composition Lemma

Subsequently, the Gröbner basis theory has been developed for other types of algebras, such as associative algebras by Bokut in [Bok76] and by Bergman in [Ber78]. They prove Newman’s Lemma for rewriting systems in free associative algebras compatible with a monomial order stating that local confluence and confluence are equivalent properties. This result was called Composition Lemma by Bokut and Diamond Lemma for ring theory by Bergman, see also [Mor94, Ufn95]. In general, the Buchberger algorithm does not terminate for ideals in a noncommutative polynomial ring Khxi. Indeed, its termination would give a decision procedure of the undecidable word problem. Even if the ideal is finitely generated it may not have a finite Gröbner basis. However, when K is a field an infinite Gröbner basis can be computed, [Mor94, Ufn98]. Note that ideas in the spirit of the Gröbner basis approach appear in several others works. Let us mention works by Hironaka in [Hir64] and Grauert in [Gra72] that compute bases of ideals in rings of power series having analogous properties to Gröbner bases but without a constructive method for computing such bases. In [Coh65], Cohn gave a method to decide the word problem by a normal form algorithm based on a confluence property. Finally, Janet [Jan20], Thomas [Tho37] and Pommaret [Pom78] developed the notion of involutive bases that are particular cases of Gröbner bases in the context of partial differential algebra. We refer the reader to [IM19] for an historical account on involutives bases and their applications to algebraic analysis of linear partial differential systems. Much more recently, Gröbner basis theory was developed in various noncommutative contexts such as Weyl algebras, see [SST00], or operads [DK10].

6.1. COMPOSITION LEMMA

6.1.1. Compositions in free Lie algebras. Shirshov introduced in [Shi62] an algorithm to compute a linear basis of a Lie algebra defined by generators and relations. He used the notion of composition of elements in a free Lie algebra, that corresponds to the notion of S-polynomial in the work of Buch- berger, [Buc65]. This work remained unknown outside the USSR and the two theories were developed in parallel. The algorithm completes a given set of elements in a free algebra by adding all nontrival compositions. This algorithm corresponds to the completion algorithm given by Knuth-Bendix for term rewriting systems, [KB70], and by Buchberger for commutative polynomials, [Buc65]. The Shirshov completion constructs a set, that may be infinite, such that every composition of its elements is trivial. Such a subset is called a Lie Gröbner-Shirshov basis. The key result in [Shi62] states that the set of irreducible elements for a Gröbner-Shirshov basis S forms a linear basis of the Lie algebra with defining relations S. This result is called now the Composition-Diamond Lemma for Lie algebras. For a recent account of the theory of Gröbner-Shirshov we refer the reader to [BC14]. In this subsection we summarize without proofs an analogue of Shirshov’s composition-diamond lemma for associative algebras given by Bokut in [Bok76].

` ` 6.1.2. Compositions. Let Λ1 be a free algebra over a set Λ1 and let ≺ be a monomial order on Λ1. Bokut introduced in [Bok76] the notion of composition of elements of a free associative algebra as ` ∗ follows. Given two 1-cells f and g in Λ1 and a monomial w in Λ1. There are two kinds of compositions:

∗ i) if w = lm(f)v = u lm(g) with `(lm(f)) + `(lm(g)) > `(w), for some monomials u and v in Λ1,

96 6.1.3. Gröbner-Shirshov’s bases

then the 1-cell 1 1 (f, g) = fv − ug w lc(f) lc(g) is called the intersection composition of f and g with respect to w.

∗ ii) if w = lm(f) = u lm(g)v, for some monomials u and v in Λ1, then the 1-cell 1 1 (f, g) = f − ugv w lc(f) lc(g)

is called the inclusion composition of f and g with respect to w.

A composition (f, g)w can also be called an S-polynomial of f and g with respect to w. A composition (f, g)w is either zero or satisfy (f, g)w ≺ w. Moreover the composition (f, g)w is in the ideal h f, gi generated by f and g. Note that a composition (f, g)w depends on the two polynomials f and g as well as the monomial w. Indeed, in some cases two polynomials f and g may overlap with different combinations creating several compositions.

` ∗ 6.1.3. Gröbner-Shirshov’s bases. Let G be a set of nonzero 1-cells in Λ1. Given a monomial w in Λ1, a 1-cell h is trivial modulo (G, w) if there exists a decomposition

h = λiuigivi, i∈I X ∗ with λi in K, ui, vi in Λ1 and gi in G such that ui lm(gi)vi ≺ w. ` A set G set of nonzero 1-cells in Λ1 is a Gröbner-Shirshov’s basis, GS basis for short, with respect to the monomial ordering ≺ if every composition (f, g)w of 1-cells in G is trivial modulo (G, w).A GS-basis G is minimal if there is no inclusion composition with elements of G. A minimal GS-basis G is called closed under composition in [Bok76]. A GS-basis G is reduced if the set G is reduced with respect to the monomial order ≺.

` 6.1.4. Exercise. Let G be a minimal Gröbner-Shirshov basis in a free algebra Λ1. Suppose that there exists a decomposition w = u1 lm(g1)v1 = u2 lm(g2)v2, ∗ with u1, v1, u2, v2 ∈ Λ1 and g1, g2 ∈ G. Show that u1g1v1 − u2g2v2 is trivial modulo (G, w). ` 6.1.5. Theorem (The Composition Lemma, [Bok76, Proposition 1 & Corollary 1]). Let Λ1 be a free ` ` algebra and let ≺ be a monomial order on Λ1. Let G be a set of 1-cells in Λ1 and let I be the ideal ` generated by G. Denote by A the algebra given by the quotient of the free algebra Λ1 by the ideal I. The following conditions are equivalent.

i) G is a GS-basis.

∗ ii) For any f in I, there exists a decomposition lm(f) = u lm(g)v for some u, v in Λ1 and g in G. iii) The set of G-reduced monomial forms a linear basis of the algebra A.

97 6.2. Reduction operators

6.2. REDUCTION OPERATORS

Yet another approach of rewriting in associative algebras were developed by Bergman in [Ber78]. With a functional description of linear rewriting reductions he obtained an equivalent result of the composition lemma 6.1.5.

` 6.2.1. Reduction operators. Given Λ1 a free algebra over a set Λ1, he deﬁnes a reduction system as a ` ` set S of pairs σ = (wσ, fσ), where wσ is a monomial of Λ1 and fσ is a 1-cell of Λ1. Given σ in S and ∗ ` ` two monomials u, v in Λ1, he considers the linear map ruσv : Λ1 − Λ1 deﬁned by

ufσv if w = uwσ→v, ruσv(w) = w otherwise.

The endomorphism ruσv is called reduction by σ. Note that this notion of reduction corresponds to the notion of rewriting step given in (5.2.4). ` A 1-cell f in Σ1 is irreducible under S if every reduction by elements of S acts trivially on f, that ∗ is uwσv is not in the support of f, for any σ in S and monomials u, v in Σ1. As in the case of linear nf ` ` 2-polygraphs, we denote by Λ1 the vector subspace of Λ1 of all irreducible 1-cells of Λ1.

6.2.2. Reduction-unique. Bergman introduced the notion of confluence for reduction systems as follows. A finite sequence of reductions r1, . . . , rn is final on a 1-cell f, if the 1-cell rn . . . r1(f) is irre- ` ducible. A 1-cell f of Λ1 is reduction-finite if for any infinite sequence (rn)n>1 of reductions, ri acts trivially on ri−1 . . . r1(f) for a sufficiently large i.A 1-cell f is reduction-unique if it is reduction-finite and if its images under all final sequences of reduction are the same. This common image is denoted ` by rS(f). A reduction system S is reduction-unique if all 1-cells of Λ1 are reduction-unique under S.

6.2.3. Exercise, [Ber78, Lemma 1.1.].

` ` ru 1) Show that the set of reduction-unique 1-cells of Λ1 forms a subspace of Λ1 denoted by Λ1 and that ru irr rS : Λ1 Λ1 deﬁnes a linear map.

2) Given monomials→ wf, wg and wh in the support of the 1-cells f, g and h respectively, such that the ru product wfwgwh is in Λ1 . Show that for any ﬁnite composition of reductions r, then fr(g)h is in ru Λ1 and that rS(fr(g)h) = rS(fgh) holds.

∗ 6.2.4. Ambiguities. A 5-tuple (σ, τ, u, v, w) with σ, τ in S and u, v, w monomials in Λ1, such that wσ = uv and wτ = vw (resp. σ 6= τ, wσ = v and wτ = uvw) is an overlap ambiguity (resp. inclusion ambiguity) of S. Such an ambiguity is resolvable if there exist compositions of reductions r and r0 that satisfy the conﬂuence condition:

0 0 r(fσw) = r (ufτ) resp. r(ufσw) = r (fτ) .

98 6.2.5. Reduction system compatible with a monomial order

6.2.5. Reduction system compatible with a monomial order. The diamond lemma obtained by Bergman concern reduction systems compatible with a monomial order. A reduction system S is compatible with a monomial order ≺, if for any σ = (wσ, fσ) in S, we have w ≺ wσ for any monomial w in the support of fσ. ∗ Given a reduction system compatible with a monomial order ≺. For a monomial w in Σ1, we denote ` by I≺w the subspace of Λ1 deﬁned by I = u(w − f )v | (w , f ) ∈ S uw v ≺ w . ≺w SpanK σ σ σ σ and σ An overlap ambiguity (resp. inclusion ambiguity) (σ, τ, u, v, w) is resolvable relative to ≺ if fσw − ufτ ∈ I≺uvw, resp. ufσw − fτ ∈ I≺uvw . ` ` Let G be a subset of 1-cells of Λ1 and let ≺ be a monomial order on Λ1. We denote by S(G, ≺) the reduction system generated by G with respect to ≺ deﬁned by 1 S(G, ≺) = { (lm(f), lm(f) − f) | f ∈ G }. lc(f) 6.2.6. Theorem (The Diamond Lemma, [Ber78, Theorem 1.2]). Let S be a reduction system compatible with a monomial order ≺. The following conditions are equivalent. i) All the ambiguities of S are resolvable. ii) All the ambiguities of S are resolvable relative to ≺. iii) S is reduction-unique. A fourth equivalent condition is given in [Ber78, Theorem 1.2] as follows. Consider the algebra A ` given as the quotient of the free algebra Λ1 by the two-side ideal

I(S) = { wσ − fσ | σ ∈ S }. If the reduction system S is compatible with a monomial order ≺, the conﬂuence conditions i) - iii) above irm hold if and only if the set Λ1 of irreducible monomial under S is a linear basis of the algebra A. In this nf case, the K-algebra A is isomorphic to the K-algebra Λ1 , whose product is given by f · g = rS(fg), for nf any 1-cells f and g in Λ1 .

6.3. NONCOMMUTATIVE GRÖBNERBASES

` 6.3.1. Noncommutative Gröbner bases. Let Λ1 be a free algebra over a set Λ1 and let ≺ be a mono- ` ` mial order on Λ1.A (noncommutative) Gröbner basis of an ideal I of Λ1 with respect to the monomial order ≺ is a subset G of I such that the ideal generated by the leading monomials of the 1-cells of I coincides with the ideal generated by the leading monomials of the 1-cells of G: h lm(I) i = h lm(G) i. Equivalently, for every 1-cell f in I, there exists g in G with lm(f) = u lm(g)v, where u and v are ` monomials of Λ1. The two following results show that the notion of noncommutative Gröbner basis corresponds to the notion of left-monomial convergent linear 2-polygraph compatible with a monomial order.

99 6.3. Noncommutative Gröbner bases

6.3.2. Proposition. Let Λ be a convergent left-monomial linear 2-polygraph, compatible with a mono- ` mial order ≺ on Λ1. The set of 1-cells {s1(α) − t1(α) | α ∈ Λ2} is a Gröbner basis of the ideal I(Λ) for the monomial order ≺.

6.3.3. Exercise. Prove Proposition 6.3.2.

` 6.3.4. Proposition. Let I be an ideal of a free 1-algebra Λ1. Let G be a Gröbner basis for I with respect to a monomial order ≺. Then the leading 2-polygraph Λ(G, ≺) is convergent and I(Λ(G, ≺)) = I holds.

Proof. Suppose that G is a Gröbner basis of the ideal I with respect to ≺. By deﬁnition, the ideal I(Λ(G, ≺)) is equal to the ideal I generated by G. Prove that the linear 2-polygraph Λ(G, ≺) is convergent. Its termination is a consequence of its compatibility with the monomial order ≺. The monomials ∗ in Λ1 reduced with respect to Λ(G, ≺) are the monomials that cannot be decomposed as u lm(g)v with g ∗ ` in G and u and v monomials in Λ1. As a consequence, if a reduced 1-cell f of Λ1 is contained in the ideal I, its leading monomial must be 0, because G is a Gröbner basis of I. By Proposition 5.5.4, we deduce that the linear 2-polygraph Λ(G, ≺) is conﬂuent.

As a conclusion to this chapter, the following result summarizes all the characterizations of the conﬂuence property of linear rewriting systems. Note that some equivalences are tautological.

` ` 6.3.5. Theorem. Let Λ1 be a free algebra over a set Λ1. Let ≺ be a monomial order on Λ1. Given an ` ideal I of Λ1 and a subset G of I, we denote by Λ the leading polygraph Λ(G, ≺) and by S the reduction system S(G, ≺). The following conditions are equivalent.

i) G is a Gröbner basis with respect to ≺.

ii) Λ is convergent.

iii) Λ is conﬂuent.

iv) Λ is locally conﬂuent.

v) All the critical branchings of Λ are conﬂuent.

vi) Every composition (f, g)w is reduced to 0 with respect to the division by G.

vii) All the ambiguities of S are resolvable. viii) All the ambiguities of S are resolvable relative to ≺.

ix) S is reduction-unique.

` nf x) Λ1 = Λ1 ⊕ I.

xi) Every 1-cell of I admits 0 as a normal form with respect to Λ2.

∗ xii) For any f in I, there exists a decomposition lm(f) = u lm(g)v for some u, v in Λ1 and g in G. xiii) The set of G-reduced monomials forms a linear basis of the algebra A.

100 6.3.6. Exercise

6.3.6. Exercise. Prove the equivalences of Theorem 6.3.5.

6.3.7. Example. Consider the linear 2-polygraph Λ given in Example 5.1.7. For the deglex order 3 3 3 ≺deglex induced by the alphabetic order x ≺ y ≺ z, the leading monomial of f = z + y + x − xyz is z3, so that 3 αf 3 3 Λ({f}, ≺deglex) = h x, y, z | z xyz − x − y i

The left-monomial linear 2-polygraph Λ({f}, ≺deglex) is compatible with the monomial order ≺deglex, hence it is terminating. It is not conﬂuent, because neither⇒ of its two critical branchings is conﬂuent:

2 3 3 αfz *> xyz − x z − y z

zαf 4 zxyz − zx3 − zy3

3 2 3 2 2 3 3 2 3 2 xyαf − x z − y z 4 3 3 2 3 2 αfz (< xyz − x z − y z %9 xyxyz − xy − xyx − x z − y z

2 z2xyz − z2x3 − z2y3 z αf "6

In particular, {f} does not form a Gröbner basis of the ideal I(Λ) We add to the polygraph Λ({f}, ≺deglex) the following 2-cell β : zy3 zxyz − zx3 + y3z + x3z − xyz2. This new rule makes the two previous critical branchings conﬂuent and create a new critical branching ⇒ 2 z β *> z3xyz − z3x3 + z2y3z + z2x3z − z2xyz2

z3y3

xyzy3 − x3y3 − y6 αy3 &: which is also conﬂuent. Finally, the convergent linear 2-polygraph h x, y, z | αf, β i is Tietze equivalent to the initial linear 2-polygraph Λ({f}, ≺deglex). In particular, the set of 1-cells {f, s1(β) − t1(β)} forms a Gröbner basis of the ideal I(Λ) with respect to the order ≺deglex.

6.3.8. Example. The algebra presented by the following linear 2-polygraph h x, y, z | x2 = 0, xy = zx i does not have a ﬁnite Gröbner bases on 3-generators x, y and z. Indeed, the ﬁrst relation is oriented as 2 n x 0 and the orientation xy zx induce the addition of the 2-cells xz x 0, for all integer n > 1. Another way is to orient the relation as zx xy. But in this case, we need to add the 2-cells xynx 0, for⇒ all integer n > 1. ⇒ ⇒ ⇒ ⇒

101 6.3. Noncommutative Gröbner bases

6.3.9. Exercise. Show that we can compute a Gröbner bases for the algebra given in Example 6.3.8 with four generators. [Hint. Add a generator t and the relations xy t and zx t.]

⇒ ⇒

102 CHAPTER 7

Anick’s resolution

Contents 7.1 Homology of an algebra ...... 104 7.2 Anick’s chains ...... 105 7.3 Anick’s resolution ...... 107 7.4 Computing homology with Anick’s resolution ...... 113 7.5 Minimality of Anick’s resolution ...... 115

In two seminal papers, Anick introduced a method to compute a free resolution for an algebra starting with a Gröbner basis of its ideal of relations. First he gave the construction for monomial algebras in [Ani85] then for associative augmented algebras in [Ani86]. For an algebra presented by a Gröbner basis, the nth chains of its Anick resolution are generated by the n-fold overlaps of the leading terms of the Gröbner basis, and the differentials are constructed by Noetherian induction. The chains deﬁned by Anick are recall in Subsection 7.2. The construction of the resolution is given in Subsection 7.3. Resolutions for path algebras using the same method were obtained by Anick and Green in [AG87]. For a deeper discussion on the theory of Gröbner bases for path algebras and how to apply this theory to the construction of free resolutions for path algebras, we refer the reader to [Gre99]. Let us mention that the Anick resolution has been achieved by other methods. In particular, the Anick resolution for a homogeneous algebra can be constructed by a deformation of the resolution computed on the associated monomial algebra, see [DK09, Sec. 2.4.] for details, see also the Backelin construction in [Bac78]. The Anick resolution can be also obtained using algebraic Morse theory with a Morse matching on the bar resolution, see [Skö06, Sec. 3.2.] for details. Morse theory allows to construct, starting from a chain

103 7.1. Homology of an algebra complex, a new chain complex such that the homology of the two complexes coincides. This method was also applied to the computation of minimal resolutions starting from the Anick resolution, [JW09]. Note also that others constructions of free resolutions using convergent rewriting systems were obtained by several authors, [Bro92, Kob90, Gro90, Kob05, GM12b]. Finally, let us mention that noncommutative Gröbner bases where developed by Dotsenko and Khoroshkin for shuffle operads in [DK10], giving operadic versions of Newman’s lemma and Buchberger’s algorithm. The Anick resolution for shuffle operads was constructed by Dotsenko and Khoroshkin in [DK09, DK12]. Using this construction, they prove that a shuffle operad with a quadratic Gröbner basis is Koszul, [DK12].

7.1. HOMOLOGYOFANALGEBRA

In this section, we briefly recall the definition of homology of associative algebras with coefficients in left modules. For a deeper discussion on basic notions of homological algebra we refer the reader to [HS97, Rot09].

7.1.1. Functor Tor. Let us recall the deﬁnition of the derived functor TorR of the tensor product of modules over a ﬁxed ring R. Let M be a left R-module and N be a right R-module. Given a projective resolution P of the right R-module N:

dn−1 d0 ε P : ··· / Pn / Pn−1 / ··· / P1 / P0 / N / 0 we associate the deleted complex:

dn−1 d0 PN : ··· / Pn / Pn−1 / ··· / P1 / P0 / 0 obtained by suppressing the module N. Note that, we have not lost any information in the complex PN, as N = coker(d0) by exactness of complex P. Then, applying the functor − ⊗R M, we form a complex of Z-modules, denoted by PN ⊗R M:

dn−1 d0 PN ⊗R M : ··· / Pn ⊗R M / Pn−1 ⊗R M / ··· / P1 ⊗R M / P0 ⊗R M / 0 where dn−1 denotes the map dn−1 ⊗ IdM. R For a natural number n > 0, we deﬁned the Z-module Torn(M, N) as the nth homology group of this complex: R Torn(N, M) = Hn(PN ⊗R M) = Ker dn−1/Im dn. R This deﬁnition is functorial in each variables, giving a bifunctor Torn from R-modules with values in the category of Z-modules.

R 7.1.2. Following the deﬁnition, the functor Tor0 (N, −) is naturally equivalent to N⊗R − and the functor R R Torn(−,M) is naturally equivalent to −⊗R M. Indeed, we have Tor0 (N, M) = coker(d0). Furthermore, the functor N ⊗R − is right exact, hence

coker(d0) = P0 ⊗R M/Im (d0) = P0 ⊗R M/ ker(ε ⊗ IdM) = N ⊗R M. This proves that R Tor0 (N, M) = N ⊗R M.

104 7.1.3. Contracting homotopy

7.1.3. Contracting homotopy. Recall that a method to prove that a complex of R-modules

dn dn−1 d0 ε ··· / Mn+1 / Mn / Mn−1 / ··· / M1 / M0 / N / 0 is acyclic is to construct a contracting homotopy, that is a sequence of morphisms of abelian groups

in+1 in i1 i0 ··· o Mn+1 o Mn o Mn−1 o ··· o M1 o M0 o N such that

ει0 = IdN, d0ι1 + ι0ε = IdM0 , dnιn+1 + ιndn−1 = IdMn , for every n > 1.

7.1.4. Homology of an algebra. Let A be an associative algebra over a field K. For n > 0, the n-th homology space of the algebra A with coefficient in a left A-module M is defined by

A Hn(A,M) = Torn(K,M).

In practice, to compute the n-th homology spaces Hn(A, K), for all n > 0, we construct a free resolution of K, seen as a trivial right-A-module:

d d / n−1 / / / 0 / ε / / FK : ··· Fn Fn−1 ··· F1 F0 K 0 and we compute the homology of the complex FK ⊗A K.

7.2. ANICK’SCHAINS

In this subsection, Λ denotes a reduced left-monomial linear 2-polygraph. The set of sources of rules ∗ ∗ in Λ2 we will be denoted by s1(Λ) = { s1(α) ∈ Λ1 | α ∈ Λ2 }. For a monomial u in Λ1, we denote by deg (u) the number of possible reductions on u with respect to Λ . s1(Λ) 2

7.2.1. Anick’s chains, [Ani86]. For an integer n > −1, the Anick n-chains of the linear 2-polygraph Λ and their tails are deﬁned by induction as follows.

- The unique (−1)-chain is the empty monomial, denoted by 1, it is its own tail.

- The 0-chains are the 1-cells in Λ1, and the tail of a 0-chain x in Λ1 is x itself.

∗ - For n > 1, suppose that (n − 1)-chains and their tails constructed. An n-chain is a monomial u in Λ1 of the form u = vt such that

i) v is (n − 1)-chain,

105 7.2. Anick’s chains

ii) t is a reduced monomial with respect to Λ2, called the tail of u, iii) if r is the tail of v, then deg (rt) = 1, s1(Λ)

iv) the unique reduction on rt is rightmost, that is, given by a 2-cell α in Λ2 reducing the ending of the monomial rt:

u / v / t / / r F α Õ

We will denote by Ωn(Λ), or by Ωn if there is no possible confusion, the set of n-chains of the linear 2-polygraph Λ.

7.2.2. Anick’s chains and overlapping. The linear 2-polygraph Λ being reduced, we have the following description of Anick’s chains. We have Ω1(Λ) = s1(Λ). Indeed, a 1-chain is a non reduced monomial u written u = xt1, where x is a 1-cell in Λ1 and t1 is a reduced monomial:

u / t x / 1 / B

Õ and such that there is only one 2-cell in Λ2 that can be applied on the monomial u.A 2-chain u is the source of a critical branching. Indeed, u = xt1t2, where xt1 is the source of a 2-cell α in Λ2 and there is a rightmost reduction τ reducing t1t2, and thus overlapping α:

t t x / 1 / 2 / B B α Õ τ Õ

Moreover, u is not the source of a critical triple branching, as we have deg u = 2. In this way, there s1(Λ) is a one-to-one correspondence between Ω2(Λ) and the set of critical branchings of the 2-polygraph Λ. For n > 3, a n-chain u corresponds to a n-fold overlapping composed by (n − 1) chained critical branchings. It is possible that deg u > n, see Example 7.2.5. s1(Λ)

x t1 t2 t3 t4 t5 ··· B B B B B Õ Õ Õ Õ Õ

106 7.2.4. Notation

7.2.3. Proposition ([Ani86]). Suppose n > 1. If u = xi1 . . . xit is an n-chain, then there is a unique s 6 t such that xi1 . . . xis is an (n − 1)-chain. Moreover, xis+1 . . . xit is reduced. (n − 1) x . . . x x . . . x u Indeed, suppose that there is two -chains i1 is and i1 is 0 which factorise . By uniqueness of the reduction on the tail, condition iii) in (7.2.1), necessarily we have s = s0

7.2.4. Notation. An n-chain u, whose (n − 1)-chain is v and tail is t, will be denoted by u = v|t. Expanding this notation, any n-chain can be written x|t1|t2| ... |tn, where x ∈ s1(Λ) and x|t1| ... |ti is an i-chain for any 0 < i < n.

7.2.5. Example, [Ani86]. Let Λ be a reduced left-monomial linear 2-polygraph with Λ1 = {x} and 3 3 2 s1(Λ) = {x }. The 1-cell x is the unique 0-chain. The monomial x = x|x is the unique 1-chain, xx is not a 1-chain because deg x2 = 0. The monomial x4 = x3|x is the unique 2-chain. Note that s1(Λ) x5 = x3|x2 is not a 2-chain. Indeed, deg x4 = 2, and on the monomial x5 there are three possible s1(Λ) reductions, with the ﬁrst one that intersects the last one, giving a critical triple branching: xxxxx The monomial x6 = x4|x2 is the unique 3-chain. Note that x5 = x4x is not a 3-chain because deg xx = 0. Note that there are four possible reductions on the 3-chain x6: s1(Λ) xxxxxx Thus we have 4 6 Ω0 = Λ1,Ω1 = s1(Λ),Ω2 = {x },Ω3 = {x }. More generally, we show that for any integer n > 0, we have 3n 3n+1 Ω2n−1 = {x },Ω2n = {x }.

2 2 7.2.6. Example, [Ani86]. Suppose that Λ1 = {x, y} and s1(Λ) = {x yxy, xyxy }. Then we have 2 2 Ω0 = {x, y},Ω1 = {x|xyxy, x|yxy },Ω2 = {x|xyxy|y, x|xyxy|xy },Ωn = ∅, for n > 3.

7.2.7. Exercise, [Ani85]. Let Λ be a linear 2-polygraph such that Λ1 = {x, y, z}. Determine Anick’s chains in the following situations

1) s1(Λ) = {xyzx, zxy},

2) s1(Λ) = {xyzx, xxy}. In this case, show that the number of n-chains equals the (n + 2)nd Fibonacci number when n > 1.

7.3. ANICK’SRESOLUTION

In this subsection, Λ denotes a convergent reduced left-monomial linear 2-polygraph, whose 2-cells are ∗ compatible with a monomial order ≺ deﬁned on Λ1. Let denote by A the algebra presented by Λ. We ` ` deﬁne a section ι : A − Λ1 of the canonical projection π : Λ1 − A, sending every 1-cell f of A to ` the normal form fbof any representative 1-cell of f in Λ1, as in (5.5.5). → →

107 7.3. Anick’s resolution

7.3.1. Anick’s resolution. Let A[Ωn(Λ)] = K[Ωn(Λ)] ⊗K A be the free right A-module over the set of n-chains Ωn(Λ). We will identify A[Ω0(Λ)] to A[Λ1] and A[Ω−1(Λ)] to A. Anick constructs in [Ani86] a free resolution of right A-modules deﬁned by the complex

dn d1 d0 ε A(Λ): · · · − A[Ωn(Λ)] A[Ωn−1(Λ)] · · · − A[Ω1(Λ)] A[Λ1] A K − 0,

d whose differentials→ n are constructed→ inductively→ simultaneously→ with→ the contracting→ homotopy→ →

ιn : Ker dn−1 − A[Ωn(Λ)].

d ι The applications n are morphisms of right A-modules→ and the applications n are linear maps.

7.3.2. For the ﬁrst steps of the resolution

d 0 / ε / A[Λ1] o A o K / 0, (7.3.3) ι0 ι−1 we deﬁne ι−1 : K , A as the embedding of K in A, and we deﬁne the augmentation map ε : A K by setting ε(x) = 0, for all x ∈ Λ1. Hence, we have A = K ⊕ Ker ε, and ει−1 = IdK. Then, we set → → d0(x ⊗ 1) = x,

∗ for all x in Λ1. By convergence hypothesis, any monomial in A admits a unique normal form in Λ1 with ∗ respect to Λ2. For a monomial u in A such that the normal form is written ub = x1x2 . . . xk in Λ1, we deﬁne ι0(1 ⊗ u) = x1 ⊗ x2 . . . xk. (7.3.4)

Then, we extend ι0 to any f in A by linearity. The map ι0 is well deﬁned by uniqueness of the normal form due to the convergence of the linear 2-polygraph Λ. The exactness of the sequence (7.3.3) in A is a consequence of the two equalities:

εd0(x ⊗ 1) = 0 and d0ι0 = idKer (ε).

7.3.5. For n > 1, we deﬁne the pair (dn, ιn) by induction on n:

d d d n / n−1 / n−2 / A[Ωn(Λ)] o A[Ωn−1(Λ)] o A[Ωn−2(Λ)] o ··· ιn ιn−1 ιn−2

We suppose that the maps dk and ιk : Ker dk−1 − A[Ωk(Λ)] are constructed such that

d d = 0 and d ι = Id , k−1 k → k k Ker dk−1 for all k 6 n − 1. We deﬁne inductively dn on an n-chain v|t with tail t by

dn(v|t ⊗ 1) = v ⊗ t − ιn−1dn−1(v ⊗ t). (7.3.6)

108 7.3.7.

7.3.7. In the deﬁnition of dn(v|t ⊗ 1), the term v ⊗ t will be the leading term with respect the well- ` founded order deﬁned on A[Ωn(Λ)] as follows. We extend the monomial order ≺ on Λ1 into a wellfounded order on A[Ωn(Λ)] by setting

f1 ⊗ u1 ≺ f2 ⊗ u2 if f1ub1 ≺ f2ub2, for all f1, f2 in K[Ωn(Λ)] and u1, u2 in A.

7.3.8. Let us deﬁne recursively the map

ιn : Ker dn−1 − A[Ωn(Λ)] as follows. Given h in Ker d ⊂ A[Ω (Λ)], we denote by u ⊗ t the leading term of h, that is n−1 n−1 → n−1

h = λun−1 ⊗ t + (lower terms), where λ in K is non-zero. The (n − 1)-chain un−1 can be uniquely decomposed in

0 un−1 = un−2|t ,

0 where un−2 is an (n − 2)-chain and t is the tail of un−1. By induction, we have

0 dn−1(un−1 ⊗ 1) = un−2 ⊗ t + (lower terms).

As dn−1 is a morphism of right A-modules, we have

dn−1(h) = λdn−1(un−1 ⊗ t) + dn−1(lower terms) 0 = λun−2 ⊗ t t + (lower terms).

0 0 Suppose now that the monomial t t is reduced, then un−2 ⊗ t t remain the leading term of dn−1(h), 0 hence h cannot be in Ker dn−1 thus contradicting the hypothesis. It follows that t t can be reduced, and we set t0t = v0wv, 0 where w is the 1-source of the leftmost reduction with respect to Λ2 that can be applied on t t:

un−1 un−2 t0 t (7.3.9) v0 w v w2 w1

0 Consider the factorization w = w2w1 and t = w1v as in the picture (7.3.9). It follows that un−2v w = 0 0 un−2|t |w1 forms an n-chain, and un−2v w ⊗ v ∈ A[Ωn(Λ)]. We set

ιn(h) = ιn(λun−1 ⊗ t + lower terms) 0 0 = λun−2v w ⊗ v + ιn(h − λdn(un−2v w ⊗ v)).

109 7.3. Anick’s resolution

0 This is well deﬁned, because h − λdn(un−2v w ⊗ v) ≺ h by construction. Indeed

0 0 0 dn(un−2v w ⊗ v) = dn(un−2v w2w1 ⊗ v) = un−2v w2 ⊗ w1v + (lower terms)

= un−1 ⊗ t + (lower terms).

0 Moreover, dn−1(h − λdn(un−2v w ⊗ v)) = 0. From this construction, we deduce the following result:

7.3.10. Theorem ([Ani86, Theorem 1.4]). Let A be an algebra presented by a convergent reduced left- monomial linear 2-polygraph Λ, compatible with a monomial order ≺. The complex of right A-modules A(Λ) deﬁned by

dn d1 d0 ε · · · − A[Ωn(Λ)] A[Ωn−1(Λ)] · · · − A[Ω1(Λ)] A[Λ1] A K − 0 where, for any→n > 0, the morphism→ dn is deﬁned→ on→ a n-chain v|t by→ → → →

dn(v|t ⊗ 1) = v ⊗ t + h, where lt(h) ≺ v|t ⊗ 1, if h 6= 0, is a resolution of the trivial right A-module K.

7.3.11. Example. Let consider the algebra A presented by the linear 2-polygraph

α Λ = h x, y | x2 0 yx i, compatible with the deglex order ≺deglex induced by the⇒ alphabetic order y ≺ x. It appears one critical branching xα0 )= xyx x3 y2x 5I 2 α0x !5 yx yα0 We complete the linear 2-polygraph Λ with the 2-cells

n n+1 αn : xy x %9 y x, for all n > 0. We note that, for any integers n, m > 0, we have a critical branching

n n+m+1 xy αm xy x αn+m+1 )= EY 0 xynxymx αn,m yn+m+2x .B m !5 n+1 m n+1 αny x y xy x y αm

0 0 Then the linear 2-polygraph Λ , whose set of 1-cell is Λ1 and Λ2 = {αn | n > 0} is convergent, compati- n n+1 ble with the monomial order ≺ and Tietze equivalent to Λ. Equivalently, the set {xy x − y x | n > 0}

110 7.3.11. Example

n forms a Gröbner basis for the ideal I(Λ). Anick’s 1-chains are of the form x|y x with n > 0 and Anick’s n m 2-chains are of the form x|y x|y x with n, m > 0. More generally, for any k > 2, we have

n1 n2 nk Ωk = {x|y x|y x| ... |y x for n1, . . . , nk > 0},

Let us compute the boundary maps d0, d1, d2 and d3. We have d0(x ⊗ 1) = x, d0(y ⊗ 1) = y and

n n n d1(x|y x ⊗ 1) = x ⊗ y x − ι0d0(x ⊗ y x), n n = x ⊗ y x − ι0(1 ⊗ xy x), n n+1 = x ⊗ y x − ι0(1 ⊗ y x), = x ⊗ ynx − y ⊗ ynx.

The last equality is consequence of the deﬁnition of the map ι0 in (7.3.4).

n m k n m k n m k d3(x|y x|y x|y x ⊗ 1) = x|y x|y x ⊗ y x − ι2d2(x|y x|y x ⊗ y x), n m k n m k n+m+1 k = x|y x|y x ⊗ y x − ι2 x|y x ⊗ y xy x − x|y x ⊗ y x , n m k n m+k+1 n+m+1 k = x|y x|y x ⊗ y x − ι2 x|y x ⊗ y x − x|y x ⊗ y x , = x|ynx|ymx ⊗ ykx − x|ynx|ym+k+1x ⊗ 1 n m+k+1 n+m+1 k n m+k+1 − ι2 x|y x ⊗ y x − x|y x ⊗ y x − d2(x|y x|y x ⊗ 1) , = x|ynx|ymx ⊗ ykx − x|ynx|ym+k+1x ⊗ 1 n m+k+1 n+m+1 k n m+k+1 n+m+k+2 − ι2 x|y x ⊗ y x − x|y x ⊗ y x − x|y x ⊗ y x − x|y x ⊗ 1 , n m k n m+k+1 n+m+1 k n+m+k+1 = x|y x|y x ⊗ y x − x|y x|y x ⊗ 1 − ι2 − x|y x ⊗ y x − x|y x ⊗ 1 , = x|ynx|ymx ⊗ ykx − x|ynx|ym+k+1x ⊗ 1 + x|yn+m+1x|ykx ⊗ 1 n+m+1 k n+m+k+1 n+m+1 k + ι2(x|y x ⊗ y x − x|y x ⊗ 1 − d2(x|y x|y x ⊗ 1) , = x|ynx|ymx ⊗ ykx − x|ynx|ym+k+1x ⊗ 1 + x|yn+m+1x|ykx ⊗ 1 n+m+1 k n+m+k+1 n+m+1 k n+m+k+1 + ι2(x|y x ⊗ y x − x|y x ⊗ 1 − x|y x ⊗ y x − x|y x ⊗ 1) , = x|ynx|ymx ⊗ ykx − x|ynx|ym+k+1x ⊗ 1 + x|yn+m+1x|ykx ⊗ 1.

111 7.3. Anick’s resolution

7.3.12. Example. Let consider the algebra A given in 7.3.11 with the following presentation

β h x, y | yx x2 i, compatible with the deglex order induced by the alphabetic order x ≺ y. This polygraph does not have ⇒ critical branching, thus the sets of Anick’s n-chains are empty for n > 2. It follows that the associated Anick resolution is d1 d0 ε · · · − 0 − A[y|x] A[x, y] A K − 0 with d0(x ⊗ 1) = x, d0(y ⊗ 1) = y and → → → → → → d1(y|x ⊗ 1) = y ⊗ x − ι0(1 ⊗ yx), 2 = x ⊗ y − ι0(1 ⊗ x ), = x ⊗ y − x ⊗ x.

7.3.13. Example. Consider Example 5.1.7 with the algebra A presented by h x, y, z | xyz = x3 + y3 + z3 i. With the Gröbner basis computed in 6.3.7: α β z3 f xyz − x3 − y3 zy3 zxyz − zx3 + y3z + x3z − xyz2 n n 3 Anick’s chains are of the form z and z y , for n > 0, so that the Anick resolution, deﬁned in [Ani86], is inﬁnite. ⇒ ⇒

7.3.14. Exercise, [Ani86, Section 3]. Compute the Anick resolution for the algebra presented by the linear 2-polygraph h x, y | xyxyx xyx i.

7.3.15. Anick’s resolution for a monomial⇒ algebra. We construct the Anick resolution in the case of a monomial algebra A. Recall from 5.1.19, that such an algebra can be presented by a monomial linear 2-polygraph Λ, that is, left-monomial and t1(α) = 0 for all α in Λ2. Obviously, such a presentation is always convergent. Suppose that the polygraph Λ is reduced. The sets of chains for Λ are Ω0(Λ) = Λ1, Ω1(Λ) = s1(Λ) and for any n > 2, Ωn(Λ) is the set of n-overlapping x|t1| ... |tn−1|tn of branchings ∗ of Λ with x ∈ Λ1, and t1, . . . , tn ∈ Λ1, such that xt1, titi+1 in s1(Λ) for any 1 6 i 6 n − 1. We have

xtc1 = 0 and t\i−1ti = 0, for all 1 6 i 6 n. (7.3.16) Consider the boundary map dn : A[Ωn(Λ)] − A[Ωn−1(Λ)] deﬁned by → dn(x|t1| ... |tn−1|tn ⊗ 1) = x|t1| ... |tn−1 ⊗ tn − ιn−1dn−1(x|t1| ... |tn−1 ⊗ tn).

By deﬁnition of dn−1, we have

dn−1(x|t1| ... |tn−1 ⊗ tn) = x|t1| ... |tn−2 ⊗ tn−1tn − ιn−2dn−2(x|t1| ... |tn−2 ⊗ tn−1tn)

Using relation in (7.3.16), we have dn−1(x|t1| ... |tn−1 ⊗ tn) = 0, hence

dn(x|t1| ... |tn−1|tn ⊗ 1) = x|t1| ... |tn−1 ⊗ tn.

112 7.4. Computing homology with Anick’s resolution

7.4. COMPUTINGHOMOLOGYWITH ANICK’SRESOLUTION

7.4.1. Computing homology. Given an algebra A and a left A-module M. When the algebra is presented by a convergent reduced left-monomial linear 2-polygraph Λ, compatible with a monomial order, the Anick resolution A(Λ) gives a method to compute the homology groups of A with coefﬁcient in M. In this section, we give several examples of computations of homology groups with coefﬁcients in K. From the resolution A(Λ), we compute the complex A(Λ) ⊗A K given by

dn d1 d0 · · · − K[Ωn(Λ)] − K[Ωn−1(Λ)] · · · − K[Ω1(Λ)] − K[Λ1] − K − 0 where K[Ωn(Λ→)] denotes the→ free vector space→ on Ωn→(Λ) and dn denotes→ the map→ dn ⊗→IdK. These maps satisfy dndn+1 = 0, for all n > 0, and we have

H0(A, K) = K, and Hn(A, K) = Ker dn−1/Im dn.

As a ﬁrst application, we have the following ﬁniteness properties.

7.4.2. Proposition. Let A be an algebra presented by a ﬁnite convergent left-monomial linear 2-polygraph. The following statements hold.

i)A is of homological type right-FP , that is, there exists an infinite length free finitely generated resolution of the trivial right A-module K. ∞ ii) For any n > 0, the vector space Hn(A, K) is finitely generated. iii) [Ani86, Lemma 3.1] The algebra A has a Poincaré series

n PA(t) = dimK(Hn(A, K))t , ∞ n=0 X

with exponential or slower growth, that is, there are constants c1, c2 > 0, such that

n 0 6 dimK(Hn(A, K)) 6 c2(c1) .

Note that the ﬁniteness conditions i) and ii) were obtained by Kobayashi for monoids. A monoid M is of homological type right-FP over K if the monoid algebra KM is of homological type right- FP . In [Kob90], by constructing a resolution similar to the Anick resolution, Kobayashi shows that a monoid M having a presentation by∞ a ﬁnite convergent rewriting system is of homological type FP . Similar∞ constructions of resolutions of monoids presented by convergent rewriting systems were also obtained by Brown [Bro92] and by Groves [Gro90]. The diferent constructions are based on distinct∞ ways to describe the n-fold critical branchings of a convergent rewriting system.

7.4.3. Exercise. Prove the conditions i) and ii) in Proposition 7.4.2.

113 7.4. Computing homology with Anick’s resolution

7.4.4. Low-dimensional homology. In the ﬁrst dimensions, we have the following complex

d2 d1 d0 K[Ω2(Λ)] K[Ω1(Λ)] K[Λ1] K − 0

The map d0 is zero, hence → → → → H1(A, K) = K[Λ1]/Im d1.

A 1-cell x of Λ1 in Im d1 comes from a relation with source or target x. It follows that x is a redundant generator in the presentation. Indeed, a term x ⊗ 1, with x in Λ1 appears in Im d1 if and only if x is the source or the target of a 2-cell in Λ2. Let α : x y1 . . . yk be a 2-cell in Λ2, where by hypothesis y1 . . . yk is reduced. Thus we have ⇒ d1(1|x ⊗ 1) = x ⊗ 1 − ι0d0(x ⊗ 1)

= x ⊗ 1 − ι0(1 ⊗ y1 . . . yk)

= x ⊗ 1 − y1 ⊗ y2 . . . yk.

α Hence d1(x) = x. Suppose now that x1 . . . xk y is a 2-cell in Λ2. We have

d1(x1 . . . xk ⊗ 1) = x1 ⇒⊗ x2 . . . xk − ι0d0(x1 ⊗ x2 . . . xk) = x1 ⊗ x2 . . . xk − ι0(1 ⊗ y)

= x1 ⊗ x2 . . . xk − y ⊗ 1

Hence d1(x1 . . . xk) = −y. Thus, we have d1 = 0 if and only if the number of generators is minimal. In this way, dimK H1(A, K) is equal to the minimal number of generators for a presentation of the algebra A. For analogous reasons, we show that dimK H2(A, K) is the minimal required number of the deﬁning relations.

7.4.5. Example. Consider the algebra A presented by the linear 2-polygraph h x, y | yx x2 i. From the Anick resolution computed in 7.3.12, we deduce the complex ⇒ d1 d0 · · · − 0 − K[y|x] K[x, y] K − 0 whose boundary maps d0 and d1 are→ zero.→ We deduce→ → →

K if n = 0, 2, 2 Hn(A, K) = K if n = 1, 0 if n > 3.  7.4.6. Exercise [Ani86, Theorem 3.2]. Let A be an algebra admitting a presentation by a left-monomial reduced linear 2-polygraph compatible with a monomial order and having no critical branching. Show that Hn(A, K) = 0, for any n > 3. A presentation without critical branching is called combinatorially free in [Ani86].

114 7.5. Minimality of Anick’s resolution

7.5. MINIMALITYOF ANICK’SRESOLUTION

7.5.1. Minimal complex. A complex of free right A-modules

dn dn−1 · · · − Fn+1 − Fn − Fn−1 − ···

minimal d = d ⊗ : F ⊗ − F ⊗ is if all induced maps n → n IdK→ n+1→ A K → n A K are zero. A resolution is minimal if the associated complex is minimal. Note that a minimal free resolution is one in which each free module has the minimal number of generators as illustrated in→ the following example.

7.5.2. Example. Let consider the algebra A presented by the linear 2-polygraph hx, y | x yi, which is compatible with the deglex order induced by y ≺ x. The Anick resolution is ⇒ d1 d0 ε 0 − A[x|1] − A[x, y] − A − K − 0 with → → → → → d0(x ⊗ 1) = x, d0(y ⊗ 1) = y, d1(x|1 ⊗ 1) = x ⊗ 1 − 1 ⊗ y.

This resolution is not minimal because d1 6= 0. A minimal resolution for the algebra A can be constructed from the polygraph h x | ∅ i with no 2-cell.

7.5.3. Example. Let consider the algebra A presented by the linear 2-polygraph

β h x, y, z, r, s | xy α s, yz r i compatible with the deglex order induced by the alphabetic⇒ order⇒s ≺ r ≺ z ≺ y ≺ x . There is a critical branching: xyz αz xβ qÕ - szey γ xr

γ 0 which is conﬂuent by adding the rule xr sz. The linear 2-polygraph Λ = h Λ1 | α, β, γ i is compatible with the deglex order considered above, convergent and Tietze equivalent to Λ. The induced 0 the Anick resolution A(Λ ) is ⇒

d2 d1 d0 ε · · · − 0 − A[xy|z] − A[x|y, x|r, y|z] − A[x, y, z, r, s] − A − K − 0 with → → → → → → →

115 7.5. Minimality of Anick’s resolution

This resolution is not minimal, because the maps d1 and d2 are non zero. Note that

K if n = 0, 3 Hn(A, K) = K if n = 1, 0 if n > 2. and a minimal resolution for the algebra A can be constructed from the linear 2-polygraph h x, y, z | ∅ i which produces the following resolution

d0 ε · · · − 0 − A[x, y, z] A K − 0

7.5.4. Exercise. Consider the linear→2-polygraph→ → → →

β Λ = h x, y, z, r, s | xy α ss, yz sr i.

1) Complete the polygraph Λ into a convergent polygraph Λ0. ⇒ ⇒ 2) Show that the Anick resolution of Λ0 is not minimal.

3) Compute the homology of the algebra A presented by Λ.

4) Compute a minimal Anick’s resolution of the algebra A.

7.5.5. Example. Let consider the algebra

Ah a, b, c, d, e | ab = ee, bc = ed i.

The alphabetic order e ≺ d ≺ c ≺ b ≺ a induces the following orientation:

β ab α ee, bc ed.

There is only one critical branching: ⇒ ⇒ abc αc aβ rÖ - eecey γ aed

γ completed by adding the rule aed eec. The rewriting system {α, β, γ} is convergent. Anick’s chains are ⇒ Ω−1 = {1},Ω0 = {a, b, c, d, e},Ω1 = {a|b, a|ed, b|c},Ω2 = {ab|c},Ωn = ∅, for n > 3. The Anick resolution with this oriented presentation is

d2 d1 d0 ε 0 K{ab|c} ⊗ A K{a|b, a|ed, b|c} ⊗ A K{a, b, c, d, e} ⊗ A A K − 0 (7.5.6)

→ → → → → →

116 7.5.8. Exercise

with d0(x ⊗ 1) = x, for any x ∈ Ω0,

d1(a|b ⊗ 1) = a ⊗ b − e ⊗ e, d1(a|ed ⊗ 1) = a ⊗ ed − e ⊗ ec, d1(b|c ⊗ 1) = b ⊗ c − e ⊗ d, and d2(ab|c ⊗ 1) = ab ⊗ c − aed ⊗ 1. It follows that K if n = 0,  5 if n = 1, H (A, ) = K n K  2 K if n = 2,  0 if n > 3.   Hence the Anick resolution with these presentation is not minimal. A minimal Anick’s resolution for the same algebra A can be constructed with the following orientation, induced by the alphabetic order with b ≺ e ≺ a: β 0 ab α ee, ed bc which produces the following chains: ⇒ ⇒ Ω−1 = {1},Ω0 = {a, b, c, d, e},Ω1 = {a|b, e|d},Ω2 = ∅, for n > 1.

The Anick resolution with this oriention is

d1 d0 ε 0 K{a|b, e|d} ⊗ A K{a, b, c, d, e} ⊗ A A K − 0 (7.5.7)

with d0(x ⊗ 1) = x, for any x ∈ Ω0 and → → → → →

d1(a|b ⊗ 1) = a ⊗ b − e ⊗ e,

d1(e|d ⊗ 1) = e ⊗ d − b ⊗ c.

This resolution is minimal.

7.5.8. Exercise. Let consider the algebra presented by

h x, y, z, r, s | xy = ss, yz = rr i.

Show that there is no orientation of rules of this presentation giving a convergent linear 2-polygraph, and thus there is no minimal Anick’s resolution for this algebra.

7.5.9. Proposition. Let Λ be a monomial linear 2-polygraph. Let A be the monomial algebra presented by Λ. The following statements hold.

i) The Anick resolution A(Λ) deﬁned in (7.3.1) is a minimal resolution.

A ii) There is an isomorphism Torn(K, K) ' KΩn−1, for all n > 0.

117 7.5. Minimality of Anick’s resolution

Let us mention another consequence for quadratic algebras. Given a monomial linear 2-polygraph Λ which is quadratic, that is its 2-cells are of the form xixj 0, with xi, xj in Λ1. Then the Anick resolution A(Λ) is concentrated in the diagonal in the following sense. The set of 0-chains is Λ1 and they are of degree 1. The set of 1-chains is s1(Λ) and they are⇒ of degree 2. More generally, an n-chains x|t1 ... |tn−1|tn is of degree n + 1. As a consequence, we have 7.5.10. Theorem. A quadratic monomial algebra is Koszul.

7.5.11. Proposition. Let A be an algebra and let Λ be a left-monomial reduced convergent linear 2- polygraph compatible with a monomial order that presents A. If the Anick resolution A(Λ) is minimal, then, for any n > 0, we have a isomorphism of spaces

Hn(A, K) ' K[Ωn−1(Λ)].

7.5.12. Exercise. Prove Proposition 7.5.11.

7.5.13. When Anick’s resolution is minimal. We have seen in Proposition 7.5.9 that the Anick resolution A(Λ) is minimal when the presentation is monomial. Following exercise gives an other situation for which the Anick resolution is minimal.

7.5.14. Exercise. Let Λ be a left-monomial reduced linear 2-polygraph compatible with a monomial order. Suppose that Λ is convergent and quadratic, that is, any 2-cell in Λ2 is of the form xi1 xi2 yi1 yi2 with xi1 , xi2 , yi1 , yi2 in Λ1. Show that the Anick resolution A(Λ) is minimal. ⇒

7.5.15. Exercise. A linear 2-polygraph is cubical if its 2-cells are of the form xi1 xi2 xi3 yi1 yi2 yi3 . Is the result of Exercise 7.5.14 can be extended to cubical convergent linear 2-polygraphs ? ⇒ 7.5.16. Exercises. Compute homology spaces of the algebras presented by the following linear 2- polygraphs 1) h x, y | xy yx i. 2) h x, y | x2 0 i. 3) h x, y | x2 y2 i. 4) h x, y | x2 xy i. 5) h x, y | x2 xy − y2 i. 6) h x, y | xyx yxy i. ⇒ ⇒ ⇒ ⇒ ⇒ ⇒

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